# Disjointness graphs of short polygonal chains

János Pach Rényi Institute, Budapest and MIPT, Moscow<EMAIL_ADDRESS><EMAIL_ADDRESS>Supported by the National Research, Development and
Innovation Office (NKFIH) grant K-131529, ERC Advanced Grant “GeoScape,” the
Austrian Science Fund grant Z 342-N31, and by the Ministry of Education and
Science of the Russian Federation in the framework of MegaGrant No.
075-15-2019-1926. Gábor Tardos Rényi Institute, Budapest and MIPT, Moscow;
<EMAIL_ADDRESS>Supported by the ERC Synergy Grant “Dynasnet” No. 810115, the
ERC advanced grant “GeoSpace” No. 882971, the National Research, Development
and Innovation Office — NKFIH projects K-132696, SSN-135643 and by the grant
of Russian Government N 075-15-2019-1926. Géza Tóth Rényi Institute, Budapest;
<EMAIL_ADDRESS>Supported by National Research, Development and Innovation
Office, NKFIH, K-131529 and ERC Advanced Grant “GeoScape,”.

###### Abstract

The disjointness graph of a set system is a graph whose vertices are the sets,
two being connected by an edge if and only if they are disjoint. It is known
that the disjointness graph $G$ of any system of segments in the plane is
$\chi$-bounded, that is, its chromatic number $\chi(G)$ is upper bounded by a
function of its clique number $\omega(G)$.

Here we show that this statement does not remain true for systems of polygonal
chains of length $2$. We also construct systems of polygonal chains of length
$3$ such that their disjointness graphs have arbitrarily large girth and
chromatic number. In the opposite direction, we show that the class of
disjointness graphs of (possibly self-intersecting) _$2$ -way infinite_
polygonal chains of length $3$ is $\chi$-bounded: for every such graph $G$, we
have $\chi(G)\leq(\omega(G))^{3}+\omega(G).$

##  1 Introduction

Ramsey theory has many applications to other parts of mathematics and computer
science [R04], including complexity theory [MW87], approximation algorithms,
[MS85], coding [KSV05], geometric data structures [MN07], graph drawing and
representation [BEF98]. Constructing nearly optimal Ramsey graphs is a
notoriously difficult combinatorial problem [F77]. The few efficient
constructions that we have are far from optimal, but they can come in handy in
those areas where we have interesting theorems, but lack nontrivial
constructions. Here we provide two examples from combinatorial geometry, based
on two classical constructions of Erdős and Hajnal [EH66, EH68]. We close this
paper with a result pointing in the opposite direction.

For any graph $G$, let $\chi(G)$ and $\omega(G)$ denote the chromatic number
and the clique number of $G$, respectively. Clearly, we have
$\chi(G)\geq\omega(G)$, and if equality holds for every induced subgraph of
$G$, then $G$ is called a perfect graph. Following Gyárfás and Lehel [GL83],
[GL85], [G85], [G87], a class of graphs $\cal G$ is said to be $\chi$-bounded
if there is a function $f$ such that $\chi(G)\leq f(\omega(G))$ for every
$G\in{\cal G}$.

Which classes of graphs are $\chi$-bounded? Or, reversing the question, if a
graph has small clique number, how can its chromatic number be large? These
questions are related to the some of the deepest unsolved problems in graph
theory. There are two different approaches that have yielded spectacular
results in recent years.

One can investigate what kind of substructures must necessarily occur in
graphs of high chromatic number. According to Hadwiger’s conjecture [H43], if
the chromatic number of a graph is at least $t$, then it must contain a
$K_{t}$-minor. (We now know that it contains a $K_{s}$-minor with
$s>t/(\log\log t)^{18}$; cf. [P20].) Gyárfás [G75] proved that if a graph has
bounded clique number and its chromatic number is sufficiently large, then it
must contain a long induced path; see also [GHM03]. According to the (still
open) Gyárfás-Sumner conjecture [S81], the same is true for any fixed tree
instead of a path. Scott and Seymour proved that the class of graphs with no
induced odd cycle of length at least $5$ is $\chi$-bounded. For many beautiful
recent results of this kind, see the survey [SS20].

The second fruitful research direction was initiated by Asplund and Grünbaum
[AG60]: Find geometrically defined classes of graphs that are $\chi$-bounded.
Given a set $S$ of geometric objects, their intersection graph (resp.,
disjointness graph) is a graph on the vertex set $S$, in which two vertices
are connected by an edge if and only if the corresponding objects have a
nonempty intersection (resp., are disjoint). It was proved in [AG60] that the
class of intersection graphs of axis-parallel rectangles in the plane is
$\chi$-bounded (see also [ChW20]). The corresponding statement is false for
boxes in $3$ and higher dimensions [B65], and even for segments in the plane
[PKK14].

For disjointness graphs $G$ of systems of segments in the plane, we have
$\chi(G)\leq(\omega(G))^{4}.$ The same is true for systems $x$-monotone
curves, that is, for continuous curves in the plane with the property that
every vertical line intersects them in at most one point. It was shown in
[PT20] that, in this generality, the order of magnitude of this bound cannot
be improved. On the other hand, we proved [PTT17] that the class of
disjointness graphs of strings (continuous curves in the plane) is not
$\chi$-bounded. Improving our construction, Mütze, Walczak, and Wiechert
[MWW18] exhibited systems of polygonal curves consisting of three segments
such that their disjointness graphs are triangle-free ($\omega=2$), yet their
chromatic numbers can be arbitrarily large.

The above results leave open the case of polygonal curves consisting of two
segments. Our first result settles this case. A polygonal curve consisting of
$k$ segments is called a polygonal $k$-chain.

Theorem 1. There exist arrangements of polygonal $2$-chains in the plane whose
disjointness graphs are triangle-free and have arbitrarily large chromatic
numbers.

We do not know if Theorem 1 can be strengthened by requiring that the
disjointness graph of the curves has large girth.

Problem 2. _Do there exist arrangements of polygonal $2$-chains in the plane
whose disjointness graphs have arbitrarily large girth and chromatic number?_

Our next result shows that the answer to the above question is in the
affirmative if, instead of $2$-chains, we are allowed to use polygonal
$3$-chains.

Theorem 3. _For any integers $g$ and $k$, there is an arrangement of non-
selfintersecting polygonal $3$-chains in the plane whose disjointness graph
has girth at least $g$ and chromatic number at least $k$._

A _$1$ -way infinite_ polygonal $2$-chain is the union of a half-line and a
segment that share an endpoint. In our proof of Theorem 1, we actually
construct arrangements of $1$-way infinite polygonal $2$-chains whose
disjointness graphs are triangle free, but have arbitrarily large chromatic
number. Doubly tracing these $1$-way infinite $2$-chains and slightly
perturbing the resulting curve, we obtain an arrangement of _$2$ -way
infinite_ $4$-chains, _i.e.,_ $4$-chains whose first and last pieces are half-
lines. Hence, we obtain the following

Corollary 4. There exist arrangements of $2$-way infinite polygonal $4$-chains
in the plane whose disjointness graphs are triangle-free and have arbitrarily
large chromatic numbers.

Our next theorem shows that Corollary 4 is optimal: the class of disjointness
graphs of (possibly self-intersecting) $2$-way infinite polygonal $3$-chains
is $\chi$-bounded.

Theorem 5. _Let $G$ be the disjointness graph of an arrangement of $2$-way
infinite polygonal $3$-chains in the plane. Then we have
$\chi(G)\leq(\omega(G))^{3}+\omega(G)$._

In fact, we will establish Theorem 5 in a somewhat stronger setting: for
arrangements of $2$-way infinite curves that consist of three $x$-monotone
pieces; see Theorem 7. With more work, the bound in Theorem 5 and Theorem 7
can be improved to $\chi(G)\leq(\omega(G))^{3}$.

In the polygonal case, our proof is algorithmic. There is a polynomial time
algorithm in the number of the polygonal chains, which, for every $k$, either
finds $k$ pairwise disjoint chains or produces a coloring of their
disjointness graph with at most $k^{3}$ colors.

In Sections 2 and 3, we establish Theorems 1 and 3, respectively. Section 4
contains the proof of Theorem 5. We end this note with a few remarks and open
problems.

In the sequel, we informally call a polygonal $2$-chain a _V-shape_ and a
polygonal $3$-chain a _Z-shape_.

##  2 Shift graphs—Proof of Theorem 1

For every $n>1$, Erdős and Hajnal [EH68] defined the shift graph $S_{n}$, as
follows. The vertex set of $S_{n}$ consist of all pairs $(a,b)$ with $1\leq
a<b\leq n$, where two vertices, $(a,b)$ and $(a^{\prime},b^{\prime})$, are
connected by an edge if and only if $b=a^{\prime}$ or $b^{\prime}=a$. It is
easy to see that $S_{n}$ is triangle-free and that
$\chi(S_{n})=\lceil\log_{2}n\rceil$.

Order the vertices $(a,b)$ of $S_{n}$ according to the co-lexicographic order,
that is, let $(a,b)\prec(a^{\prime},b^{\prime})$ if $b<b^{\prime}$, or if
$b=b^{\prime}$ and $a<a^{\prime}$. Let $v_{1},\ldots,v_{\binom{n}{2}}$ denote
the vertices of $S_{n}$, listed in this order.

Let $v_{i}=(a,b)$ be a vertex. Its neighbors having a smaller index are
$(a^{\prime},b^{\prime})$ with $b^{\prime}=a$. No such neighbor exist if and
only if $a=1$. Notice that, for any $i$,

  1. 1.

either $v_{i}$ has no neighbor $v_{j}$ with a smaller index $j<i$,

  2. 2.

or there exist integers $c(i),d(i)$ with $1\leq c(i)\leq d(i)<i$ such that for
every $j<i$,

$v_{j}v_{i}\in E(S_{n})\Longleftrightarrow c(i)\leq j\leq d(i).$

Recall that a $1$-way infinite V-shape is the union of a half-line and a
segment that share an endpoint. In the rest of this proof, for simplicity, we
call a $1$-way infinite V-shape _long_.

Our goal is to assign a long V-shape to each vertex of $S_{n}$ so that two
V-shapes are disjoint if and only if the corresponding vertices are adjacent
in $S_{n}$. This will prove Theorem 1, because in any finite collection of
long V-shapes, we can cut the half-lines short so that the resulting (bounded)
V-shapes have the same intersection structure. Hence, we obtain a collection
of V-shapes with $S_{n}$ as its disjointness graph, and the graphs $S_{n}$ are
triangle-free and their chromatic numbers tend to infinity, as
$n\rightarrow\infty$.

We assign the long V-shape $V_{i}$ to the vertex $v_{i}$ of $S_{n}$
recursively starting at $V_{1}$. Let $h_{i}$ and $s_{i}$ denote the half-line
and the straight-line segment, respectively, comprising $V_{i}$ and let us
denote their common endpoint by $p_{i}=(x_{i},y_{i})$. We write $q_{i}$ for
the other endpoint of $s_{i}$.

During the recursive process, we will maintain the following properties:

_(i)_ $p_{i}$ is the left end point of both $h_{i}$ and $s_{i}$;

_(ii)_ both $h_{i}$ and $s_{i}$ have positive slopes;

_(iii)_ $s_{i}$ is above $h_{i}$, i.e., the slope of $s_{i}$ is larger than
the slope of $h_{i}$;

_(iv)_ for any $i>j$, the slope of $h_{i}$ will be smaller than the slope of
$h_{j}$;

_(v)_ for any $i>j$, we have $x_{i}<x_{j}$ and $y_{i}<y_{j}$.

Let $V_{1}$ be any long V-shape satisfying the above conditions. Let $i>1$,
and assume recursively that we have already constructed the long V-shapes
$V_{1},\ldots,V_{i-1}$ satisfying the above requirements. Next, we define
$V_{i}$. We distinguish two case;

Case A: The vertex $v_{i}=(a,b)$ has no neighbor with a smaller index, i.e.,
we have $a=1$.

Let $\ell$ be a horizontal line passing above $p_{1}$. It will intersect every
$V_{j}$ with $1\leq j<i$. Slightly rotate $\ell$ about any fixed point of the
plane so that the resulting line $\ell^{\prime}$ has a very small positive
slope, smaller than the slope of $h_{i-1}$ and it still intersects all $V_{j}$
for $j<i$. Choose a point $p_{i}=(x_{i},y_{i})\in\ell,$, very far to the left,
so that $x_{i}<x_{i-1}$ and $y_{i}<y_{i-1}$. Let $h_{i}$ be the part of
$\ell^{\prime}$ to the right of $p_{i}$, and let $q_{i}$ be a point to the
right of $p_{i}$ which lies above $h_{i}$. One can choose $q_{i}$ such that
the segment $s_{i}=p_{i}q_{i}$ does not intersect any of the earlier $V_{j}$.

Figure 1: Inserting $V_{i}$.

Case B: The vertex $v_{i}=(a,b)$ has at least one neighbor of smaller index,
i.e., $a>1$.

Let $c(i)$ and $d(i)$ be the constants satisfying property (2) above and let
$\ell$ be a horizontal line that passes below $p_{d(i)}$ and above
$p_{d(i)+1}$. In case $d(i)+1=i$ we could simply choose $\ell$ to be an
arbitrary horizontal line below $p_{d(i)}$, but the careful reader may notice
that this case never occurs as no vertex $v_{i}$ in $S_{n}$ is adjacent to
$v_{i-1}$.

The line $\ell$ intersects every $V_{j}$ with $d(i)<j<i$ and is disjoint from
all $V_{j}$ with $j\leq d(i)$. Slightly rotate $\ell$ about any fixed point in
the plane so that the resulting line $\ell^{\prime}$ has a very small positive
slope, smaller than that of $h_{i-1}$ and it still intersects the same
previously defined long V-shapes $V_{j}$. Select a slope $\alpha$ which is
larger than the slope of $h_{c(i)}$, but smaller than the slope of
$h_{c(i)-1}$, if $h_{c(i)-1}$ exists, that is, if $c(i)>1$.

For any $j<i$, let $\ell_{j}$ and $\ell^{\prime}_{j}$ denote the lines of
slope $\alpha$ through $p_{j}$ and $q_{j}$, respectively. Choose a point
$p_{i}=(x_{i},y_{i})\in\ell^{\prime}$ so far to the left that we have
$x_{i}<x_{i-1}$, $y_{i}<y_{i-1}$ and $p_{i}$ lies above the lines $\ell_{j}$
and $\ell^{\prime}_{j}$, for all $j\leq i$.

Let $h_{i}$ be the part of $\ell$ to the right of $p_{i}$. Let $f$ be the
half-line of slope $\alpha$, whose left endpoint is $p_{i}$. Then $f$ goes
strictly above all $s_{j}$ for $j<i$ and also of all $h_{j}$ with $c(i)\leq
j<i$, but will intersect all $h_{j}$ with $1\leq j<c(i)$. Choose $q_{i}$ on
$f$ to the right of these intersection points, then the segment
$s_{i}=p_{i}q_{i}$ also intersects all $h_{j}$ with $1\leq j<c(i)$.

Notice that the long V-shape $V_{i}$ consisting of $h_{i}$ and $s_{i}$
constructed above satisfies the conditions (i)–(v) listed above, further it
intersects exactly those other long V-shapes $V_{j}$ ($j<i$) for which $v_{j}$
and $v_{i}$ are not adjacent in $S_{n}$. See Fig. 1. This means that the
disjointness graph of the collection of the $\binom{n}{2}$ long V-shapes
constructed above is exactly $S_{n}$. This completes the proof of Theorem 1.
$\Box$

In the above proof, we have constructed a collection of $1$-way infinite
V-shapes in which each pair intersects at most twice. With a little additional
care (namely, by insisting that each $q_{i}$ is higher than $p_{1}$), we can
achieve the following. For $1\leq i<j\leq\binom{n}{2}$, with $v_{i}=(a,b)$ and
$v_{j}=(a^{\prime},b^{\prime})$, we have

  * •

if $a^{\prime}<b$, then $V_{i}$ and $V_{j}$ intersect once;

  * •

if $a^{\prime}=b$, then $V_{i}$ and $V_{j}$ are disjoint;

  * •

if $a^{\prime}>b$, then $V_{i}$ and $V_{j}$ intersect twice.

##  3 Hypergraphs of large girth—Proof of Theorem 3

A hypergraph $H$ is a pair $(V,E)$, where $V$ is a finite vertex set, $E$ is
the set of hyperedges, that is, a collection of subsets of $V$. $H$ is called
_$n$ -uniform_ if each of its hyperedges has $n$ vertices. In a _proper
coloring_ of $H$, every vertex is assigned a color in such a way that none of
the hyperedges is monochromatic. The _chromatic number_ of $H$ is the smallest
number of colors used in a proper coloring of $H$. A _Berge-cycle_ in $H$
consists of a sequence of distinct vertices $v_{1},\ldots,v_{k}$ and a
sequence of distinct hyperedges $e_{1},\ldots,e_{k}\in E$ with
$v_{i},v_{i+1}\in e_{i}$ for $1\leq i<k$ and $v_{k},v_{1}\in e_{k}$. Here $k$
is the _length_ of the Berge-cycle and it is assumed to be at least $2$. The
_girth_ of a hypergraph is the length of its shortest Berge-cycle (or infinite
if it has no Berge-cycle).

For the proof, we need the following classical result.

Erdős-Hajnal Theorem [EH66] (Corollary 13.4). _For any integers $n\geq 2$,
$g\geq 3$, and $k\geq 2$, there exists an $n$-uniform hypergraph with girth at
least $g$ and chromatic number at least $k$._

Theorem 3 is a direct consequence of part (5) of the following statement.

Lemma 6. For any integers $g\geq 3$, $k\geq 2$, there is a natural number
$n=n(g,k)$ such that for every set $P$ of $n$ points on the $x$-axis in
$\bf{R}^{2}$ and for every real $c>0$, there is an arrangement $Z=Z(P)$ of $n$
Z-shapes satisfying the following conditions.

_(1)_ Each point in $P$ is the endpoint of exactly one Z-shape in $Z$.

_(2)_ Apart from a single endpoint in $P$, every Z-shape in $Z$ lies strictly
above the $x$-axis.

_(3)_ No Z-shape in $Z$ is self-intersecting and any two cross at most twice.

_(4)_ For any Z-shape $pqrs\in Z$ with $p\in P$, whose vertices have
$x$-coordinates $x_{p}$, $x_{q}$, $x_{r}$, and $x_{s}$, respectively, we have
$x_{q}+c<x_{p}<x_{s}<x_{r}-c$.

_(5)_ The disjointness graph of the Z-shapes in $Z$ has girth at least $g$ and
chromatic number at least $k$.

Proof. For each $g$, we prove the lemma by induction on $k$. We fix $g\geq 3$.
For $k=2$, $n(g,2)=2$ is a good choice. For any two points on the $x$-axis and
any $c>0$, we can take two disjoint Z-shapes satisfying the requirements.
Their disjointness graph is $K_{2}$, its chromatic number $2$ and it has
infinite girth. See Fig. 2.

Figure 2: The case $k=2$.

Suppose now that $k\geq 2$ and that we have already proved the statement for
$k$. Now we prove it for $k+1$. Let $n=n(g,k)$.

By the Erdős-Hajnal Theorem stated above, there exists an $n$-uniform
hypergraph $H$ whose girth is at least $g$ and chromatic number at least
$k+1$. Let $v_{1},v_{2},\dots,v_{m}$ denote the vertices of $H$ and
$e_{1},e_{2},\ldots,e_{M}$ the hyperedges of $H$. Let $N=nM+m$. We show that
$n(g,k+1)=N$ satisfies the requirements of the lemma.

Let $P$ be an arbitrary set of $N$ points on the $x$-axis and let $c>0$. For
any $v_{i}\in V(H)$, let $d_{i}$ denote the _degree_ of $v_{i}$, that is, the
number of hyperedges that contain $v_{i}$. Obviously, we have

$\sum_{i=1}^{m}(d_{i}+1)=nM+m=N.$

Choose $m$ disjoint open intervals, $I_{1},\ldots,I_{m},$ such that each
$I_{i}$ contains precisely $d_{i}+1$ points of $P$. For every $i$, $1\leq
i\leq m$, we associate the interval $I_{i}$ with vertex $v_{i}$ of $H$. Let
$p_{i}$ denote the leftmost point in $P\cap I_{i}$. For every $i$ and $j$
($1\leq i\leq m$, $1\leq j\leq M$) for which $v_{i}\in e_{j}$, assign a
distinct point $p_{i}^{j}\in(P\cap I_{i})\setminus\\{p_{i}\\}$ to the pair
$(v_{i},e_{j})$.

Next, we construct a set of $N$ Z-shapes that satisfy conditions (1)–(5) of
the lemma with parameters $g,k+1$, and $c$. We construct subsets $Z_{j}$ of
our eventual set of Z-shapes for $1\leq j\leq M$. We construct these sets one
by one starting at $Z_{1}$ and using the inductive hypothesis for various
subsets of $P$ of size $n$ and with a parameter $c^{\prime}$ that we choose to
be larger than $c$ plus the diameter of $P$.

For $j=1$, consider the $n=n(g,k)$-element point set
$P^{\prime}_{1}=\\{p_{i}^{1}:v_{i}\in e_{1}\\}$. By the induction hypothesis,
there is a set $Z_{1}$ of Z-shapes such that one of their endpoints belongs to
$P^{\prime}$, and they satisfy conditions (1)–(5) with parameter $c^{\prime}$.

Suppose that $j>1$ and that we have already constructed the sets of Z-shapes
$Z_{1},\ldots,Z_{j-1}$. Now let $P^{\prime}_{j}=\\{p_{i}^{j}:v_{i}\in
e_{j}\\}$. By the induction hypothesis, there is a set $Z^{\prime}_{j}$ of
Z-shapes with one of their endpoints in $P^{\prime}$ which satisfy conditions
(1)–(5) with parameter $c^{\prime}$. Apply an affine transformation
$(x,y)\to(x,y/K_{j})$ to all Z-shapes in $Z^{\prime}_{j}$, where $K_{j}$ is a
very large constant to be specified later. The resulting family of Z-shapes,
$Z_{j}$, still satisfies all defining conditions and, by choosing $K_{j}$
large enough, we can achieve that every element of $Z_{j}$ intersects every
Z-shape in $\bigcup_{h<j}Z_{h}$ exactly once or twice.

The set $\bigcup_{j=1}^{M}Z_{j}$ contains exactly one Z-shape starting at each
point $p_{i}^{j}$. We still need to add one Z-shape
$z_{i}=p_{i}q_{i}r_{i}s_{i}$ starting at each point $p_{i},\;1\leq i\leq m$.
We define them recursively for $i=1,\ldots,m$. We make sure that each
$z_{i}=p_{i}q_{i}r_{i}s_{i}$ satisfies the following properties.

(i) _The segment $q_{i}r_{i}$ is horizontal and the $y$-coordinate of its
points is larger than the $y$-coordinate of any point of any Z-shape in
$(\bigcup_{j=1}^{M}Z_{j})\cup\\{z_{h}:1\leq h<i\\}$._

(ii) _The slope of $p_{i}q_{i}$ is $-\varepsilon_{i}$, the slope of
$r_{i}s_{i}$ is $\varepsilon_{i}$, for a sufficiently small constant
$\varepsilon_{i}>0$, to be specified later._

(iii) _The $x$-coordinate of $s_{i}$ is equal to the $x$-coordinate of the
right endpoint of $I_{i}$, and the $y$-coordinate of $s_{i}$ is
$\varepsilon_{i}$._

Clearly, if we choose $\varepsilon_{i}>0$ sufficiently small, then $z_{i}$ is
disjoint from all Z-shapes in $\bigcup_{j=1}^{M}Z_{j}$ that start in $I_{i}$,
but it intersects exactly once all other Z-shapes already defined. Also,
$z_{i}$ satisfies conditions (2) and (3), and it satisfies condition (4), too,
provided that $\varepsilon_{i}$ is sufficiently small. See Fig. 3.

Figure 3: Inserting $z_{1}$.

As we maintained conditions (1)–(4) throughout the construction, it remains
only to prove that the disjointness graph $G$ of $Z$ satisfies condition (5)
with $k+1$ in place of $k$.

To this end, let us explore the structure of $G$. The vertices of $G$ can be
partitioned into the sets $Z_{j}$ for $1\leq j\leq M$ and the independent set
$W=\\{z_{i}:1\leq i\leq m\\}$. Further, there is no edge between two distinct
sets $Z_{j}$ and $Z_{j^{\prime}}$. There is a single edge from $z_{i}$ to
$Z_{j}$ if $v_{i}\in e_{j}$, and there is no edge from $z_{i}$ to $Z_{j}$
otherwise. Finally, each vertex in $Z_{j}$ is adjacent to exactly one of the
vertices $z_{i}$, and it satisfies $v_{i}\in e_{j}$.

The structure above implies that each cycle $C$ of $G$ is either contained in
a single set $Z_{j}$, or it passes through several sets $Z_{j}$ and several
vertices in $W$. In the former case, by our assumption on the disjointness
graph of $Z_{j}$, the length of $C$ is at most $g$. In the latter case, let us
record the vertices of $W$ and the sets $Z_{j}$ as the cycle passes through
them: $z_{i_{1}},Z_{j_{1}},z_{i_{2}},Z_{j_{2}},\dots,z_{i_{h}},Z_{j_{h}}$.
Here, the vertices $v_{i_{1}},\dots,v_{i_{h}}$ are all distinct and, if the
same is true for the hyperedges $e_{j_{1}},\dots,e_{j_{h}}$, then they form a
Berge-cycle of length $h$ in the hypergraph $H$. If the hyperedges are not all
distinct, then an even shorter Berge-cycle is formed by any repetition-free
interval between two occurrences of the same hyperedge. By our assumption on
the girth of $H$, we have $h\geq g$ in both cases, so all cycles of $G$ have
length at least $g$, as required.

Suppose now that there is a proper $k$-coloring of $G$. Restricting it to the
set $W$ (and identifying each $z_{i}\in W$ with the vertex $v_{i}$ of $H$), we
obtain a $k$-coloring of the vertices of the hypergraph $H$. By our
assumption, this cannot be a proper coloring. Therefore, there is a
monochromatic hyperedge $e_{j}$. In this case, no vertex in $Z_{j}$ can
receive the common color of the vertices of $e_{j}$, so we have a proper
$(k-1)$-coloring of $Z_{j}$. This contradicts our assumption on the
disjointness graph of $Z_{j}$ and, thus, proves that $G$ has no proper
$k$-coloring. This concludes the proof of Lemma 6 and, hence, of Theorem 3.
$\Box$

James Davies [D21] used a very similar construction to show that there are
intersection graphs of axis-parallel boxes and intersection graphs of lines in
$3$-space with arbitrarily large girths and chromatic numbers.

##  4 Two-way infinite polygonal chains—Proof of Theorem 5

As we pointed out at the end of Section 2, the class of disjointness graphs of
$1$-way infinite V-shapes is not $\chi$-bounded. But if we require both ends
of a V-shape to be long, the situation will change.

A _$2$ -way infinite polygonal $k$-chain_ is a continuous curve in the plane
consisting of two half-lines connected by an (ordinary) polygonal
$(k-2)$-chain. We can relax this definition by requiring only that each of the
$k$ pieces are $x$-monotone, and the first and the last pieces have unbounded
projections to the $x$-axis. In this case, the curve is called a $2$-way
infinite _$k$ -monotone chain_.

According to this definition, a $2$-way infinite polygonal $2$-chain
(V-shape)) whose half-lines are not vertical is a $2$-way infinite _$2$
-monotone_ chain. It can also be regarded as a degenerate $2$-way infinite
_$3$ -monotone_ chain. Note that by performing a suitable rotation, if
necessary, we can always assume that none of the half-line pieces of a finite
arrangement of $2$-way infinite polygonal $k$-chains is vertical. Therefore,
the following theorem implies Theorem 5.

Figure 4: The three partial orders: $A$ is to the left of $B$, below $B$, and
above $B$.

Theorem 7. _The disjointness graph $G$ of a finite arrangement of $2$-way
infinite $3$-monotone chains satisfies
$\chi(G)\leq(\omega(G))^{3}+\omega(G)$._

Proof. We call a (possibly self-intersecting) 2-way infinite $k$-monotone
chain $A$ _wide_ if it intersects every vertical line. A chain $A$ with this
property divides the plane into (open) connected components, exactly one of
which contains a vertical half-line pointing upwards. We call this component
the _upside of $A$_. For any two wide 2-way infinite $k$-monotone chains, $A$
and $B$, we say that $A$ is _higher_ than $B$ if $A$ is contained in the
upside of $A$. In this case, the upside of $B$ is also contained in the upside
of $A$. Therefore, the relation “higher” defines a partial order on any
arrangement of wide $k$-monotone chains. According to this partial order, only
_disjoint_ pairs are comparable. Since any two disjoint _wide_ 2-way infinite
$k$-monotone chains are comparable, the disjointness graph of any collection
of wide 2-way infinite $k$-monotone chains is a comparability graph.

Now we turn our attention to the non-wide case. The complement of a non-wide
2-way infinite $k$-monotone chain $A$ has precisely one connected component
which contains a vertical line. We call this component the _large component_.
The chain $A$ is said to be a _right chain_ if $A$ is to the right of the
vertical lines in the large component, otherwise it is a _left chain_. If $A$
is a right chain, we call its large component the _left side_ of $A$. On the
other hand, if $A$ is a left chain, we call the union of _all_ connected
components of the complement of $A$, other than its large component, the _left
side_ of $A$.

For any two non-wide 2-way infinite $k$-monotone chains, $A$ and $B$, we say
that $A$ is _to the left_ of $B$ if both $A$ and its left side are contained
in the left side of $B$. Obviously, this relation also defines a partial
order, with respect to which only disjoint non-wide chains are comparable. It
is not true that any two disjoint non-wide 2-way infinite $3$-monotone chains
are comparable. Therefore, we need to introduce two further partial orders.

For any two subsets of the plane, $A$ and $B$, we say that $A$ is below $B$
($A$ is above $B$, resp.), if the following two conditions are satisfied:

  1. 1.

every vertical line that intersects $A$ also intersects $B$;

  2. 2.

if $a\in A\cap\ell$ and $b\in B\cap\ell$ for a vertical line $\ell$, then the
$y$-coordinate of $a$ is strictly _lower_ (_higher_ , resp.) than the
$y$-coordinate of $b$.

Note that “above” and “below” are two separate partial orders and not the
inverses of each other. It is clear that both of these relations are partial
orders on arbitrary planar sets and that any two comparable sets are disjoint.
See Fig. 4.

Lemma 8. _Any two disjoint non-wide 2-way infinite $3$-monotone chains, $A$
and $B$, are comparable by one of the three relations “below”, “above”, or “to
the left”._

To establish the lemma, note that non-wide 2-way infinite $3$-monotone chains
must be, in fact, 2-way infinite _$2$ -monotone_ chains. A left chain with
this property is the union of the graphs of two continuous functions
$f_{1},f_{2}:(-\infty,a]\to\mathbb{R}$, where $f_{1}(a)=f_{2}(a)$. Let $B$ be
another left chain obtained as the union of the graphs of two continuous
functions $g_{1},g_{2}:(-\infty,b]\to\mathbb{R}$, and assume that $A$ and $B$
are disjoint. We can assume, by symmetry, that $b\leq a$. Consider
$g_{1}(b)=g_{2}(b)$. It is easy to see that if it is below both $f_{1}(b)$ and
$f_{2}(b)$, then $B$ is below $A$. If it is above both $f_{1}(b)$ and
$f_{2}(b)$, then $B$ is above $A$. Finally, if $g_{1}(b)$ is between
$f_{1}(b)$ and $f_{2}(b)$, then $B$ is to the left of $A$. A similar argument
applies if both $A$ and $B$ are right chains. Finally, if a left chain is
disjoint from a right chain, then the left chain is always to the left of the
right chain. This completes the proof of Lemma 8.

Now we return to the proof of Theorem 7. Fix a family $F$ of 2-way infinite
$3$-monotone chains, and let $G$ denote their disjointness graph. Let $F_{1}$
and $F_{2}$ consist of the wide and non-wide elements of $F$, respectively. We
have seen that the disjointness graph $G[F_{1}]$ of $F_{1}$ is a comparability
graph. Comparability graphs are perfect, so we have
$\chi(G[F_{1}])=\omega(G[F_{1}])$. We also proved that the comparability graph
$G[F_{2}]$ of $F_{2}$ is the union of three comparability graphs. This implies
that $\chi(G[F_{2}])\leq(\omega(G[F_{2}]))^{3}$.

For the entire graph $G$, we have

$\chi(G)\leq\chi(G[F_{1}])+\chi(G[F_{2}])\leq\omega(G[F_{1}])+(\omega(G[F_{2}]))^{3}\leq\omega(G)+(\omega(G))^{3},$

as required. This completes the proof of the theorem. $\Box$

In [PT20], for every $k\geq 2$, we constructed arrangements of $x$-monotone
curves such that their left endpoints lie on the $y$-axis and their
disjointness graphs have clique number $k$ and chromatic number ${k+1\choose
2}.$ We can extend these curves to the left by adding horizontal half-lines
without changing their intersection structure. Traversing the resulting curves
twice, we obtain families of $2$-way infinite $2$-monotone chains such that
their disjointness graphs satisfy $\chi(G)={\omega(G)+1\choose 2}$. We do not
know whether the order of magnitude of the bound in Theorem 7 is best
possible.

We were unable to improve on the bound in Theorem 7 even for 2-way infinite
polygonal 3-chains. The best lower bound we have in this case is
$\omega(G)^{(\log 5/\log 2)-1}\approx\omega(G)^{1.32}$, and it follows from a
construction in [LMPT94].

##  5 Concluding remarks.

A. In Problem 2, we asked whether the disjointness graph of an arrangement of
V-shapes can have simultaneously arbitrarily high chromatic number and girth.
The following statement provides an affirmative answer to a relaxed version of
this question. The _odd-girth_ of a graph is the length of the shortest odd
cycle in it (or infinite if the graph is bipartite).

Proposition 9. _There exist arrangements of polygonal 2-chains in the plane
whose disjointness graphs have arbitrarily large odd-girths and chromatic
numbers._

Proof. The proof is based on the same idea as the Proof of Theorem 1, where we
represented the shift graph $S_{n}$ as the disjointness graph of an
arrangement of V-shapes. The vertices of $S_{n}$ are pairs $(a,b)$ of integers
$1\leq a<b\leq n$, so they can be associated with the edges of the complete
graph $K_{n}$. Thus, the vertices of $S_{n}$ associated with the edges of a
subgraph $G\subseteq K_{n}$ induce a subgraph $G^{*}\subseteq S_{n}$. It is
easy to verify that for any $G\subseteq K_{n}$, we have

(1) $\chi(G^{*})\geq\log(\chi(G))$ _and_

(2) _the odd-girth of $G^{*}$ is strictly larger than the odd-girth of $G$._

For any integers $g$ and $k$, there exist $n=n(g,k)$ and a subgraph $G\subset
K_{n}$ with girth (and, hence, odd-girth) at least $g$ and chromatic number at
least $k$. By properties (1) and (2), the odd-girth of the corresponding
induced subgraph $G^{*}$ of $S_{n}$ will be larger than $g$, and its chromatic
number will be at least $\log k$. The graph $G^{*}$ inherits from $S_{n}$ a
representation as a disjointness graph of V-shapes. $\Box$

Unfortunately, getting rid of short _even_ cycles, even 4-cycles, looks
impossible by using this simple trick.

B. In the example of Theorem 1 and in all interesting known examples, whenever
a class $\cal G$ of graphs, closed under taking induced subgraphs is not
$\chi$-bounded, then there exist triangle-free graphs $G\in{\cal G}$ with
arbitrary large chromatic numbers. Is this true in general? Is it true that,
for any $k$ and $\omega$, there exists $\chi=\chi(n,\omega)$ such that every
graph $G$ with $\chi(G)\geq\chi$ and $\omega(G)\leq\omega$ has an induced
triangle-free subgraph $G^{\prime}\subseteq G$ with $\chi(G^{\prime})\geq k$?
The existence of a not necessarily induced triangle-free subgraph
$G^{\prime}\subseteq G$ with this property was proved by Rödl [R77].

C. The arrangements of polygonal curves proving Theorems 1 and 3 have the
property that any two of them have at most two points in common. It would be
interesting to decide whether these theorems remain true if we insist that the
curves are _single-crossing_ , that is, any two curves have at most one point
in common at which they properly cross.

Conjecture 10. _The class of disjointness graphs of single-crossing polygonal
$2$-chains is $\chi$-bounded._

Mütze _et al._ [MWW18] proved that the same statement is false for polygonal
$3$-chains.

D. To prove Theorem 1, we established that the shift graph $S_{n}$, a
triangle-free graph of unbounded chromatic number, can be obtained as the
disjointness graph of V-shapes. However, the _fractional chromatic number_ of
$S_{n}$ is bounded: it is smaller than $4$ for every $n$. Do there exist
triangle-free disjointness graphs of V-shapes with arbitrarily large
fractional chromatic number?

Analogously, our construction for Theorem 3 gives disjointness graphs with
bounded fractional chromatic number. Do there exist disjointness graphs of
Z-shapes with arbitrarily large girth and fractional chromatic number?

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# Pressure-induced phase transition in pyrochlore iridates (Sm1-xBix)2Ir2O7
($x$ = 0, 0.02, and 0.10): Raman and X-ray diffraction studies

M Rosalin  Department of Physics, Indian Institute of Science, Bangalore,
560012, India  K.A. Irshad  Elettra-Sincrotrone Trieste S.C. p.A., S.S. 14, Km
163.5 in Area Science Park, Basovizza 34149, Italy  Boby Joseph  Elettra-
Sincrotrone Trieste S.C. p.A., S.S. 14, Km 163.5 in Area Science Park,
Basovizza 34149, Italy  Prachi Telang  Department of Physics, Indian Institute
of Science Education and Research, Pune, Maharashtra, 411008, India  Surjeet
Singh  Department of Physics, Indian Institute of Science Education and
Research, Pune, Maharashtra, 411008, India  D V S Muthu  Department of
Physics, Indian Institute of Science, Bangalore, 560012, India  A K Sood
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science,
Bangalore, 560012, India

###### Abstract

The pyrochlore iridates, $A_{2}$Ir2O7, show a wide variety of structural,
electronic, and magnetic properties controlled by the interplay of different
exchange interactions, which can be tuned by external pressure. In this work,
we report pressure-induced phase transitions at ambient temperature using
synchrotron-based X-ray diffraction (up to $\sim$ 20 GPa) and Raman-scattering
measurements (up to $\sim$ 25 GPa) of the pyrochlore series (Sm1-xBix)2Ir2O7
($x$ = 0, 0.02, and 0.10). Our Raman and X-ray data suggest an iso-structural
transition in Sm2Ir2O7 at $P_{c}$ $\sim$ 11.2 GPa associated with the
rearrangement of IrO6 octahedra in the pyrochlore lattice. The transition
pressure decreases to $\sim$ 10.2 and 9 GPa for $x$ = 0.02 and 0.10,
respectively. For all the samples, the linewidth of three phonons associated
with Ir-O-Ir ($A_{1g}$ and $E_{g}$) and Ir-O ($T_{2g}^{4}$ ) vibrations show
anomalous decrease up to $P_{c}$, due to decrease in electron-phonon
interaction.

Phase transition

##  I Introduction

Pyrochlore iridates have emerged as an interesting family of materials,
providing a unique framework for studying the interplay of electronic
correlations, spin-orbit coupling, and topological phenomena. These compounds,
$A_{2}$Ir2O7, where $A$ is a rare earth element, have drawn particular
interest due to their intricate electronic and magnetic behavior that sets the
stage for the exploration of novel quantum phenomena and topological phases
[1, 2, 3, 4, 5]. By substituting the $A$ site successively with larger rare-
earth atoms, the physical properties evolve with the $A$-site cation radius
from magnetic insulating to a complex nonmagnetic metallic behavior [6, 7, 8,
9, 10]. In the lanthanide series, from Lu to Gd, iridates show insulating
behavior ; Eu, Sm, and Nd iridates show a metal-insulator transition (MIT)
with the transition temperature varying as a function of rare earth ionic
radius [11, 12, 9]. However, the end member Pr2Ir2O7 remains metallic even at
low temperatures [10]. Increasing $A$ size in the $A$2Ir2O7 series results in
a broader Ir-O-Ir bond angle and shorter Ir-O bond lengths. As a result, the
iridium $t_{2g}$ bandwidth widens and eventually surpasses the metallization
threshold at a given $A$ ionic radius [12].

Application of chemical pressure i.e., doping $A^{3+}$ or Ir4+ sites, and
physical pressure, have proven effective tools to tune the relative scale of
interactions and electronic bandwidth in this class of materials. Expanding
the size of the $A$ cation results in a simultaneous increase in the Ir-O-Ir
bond angle and lattice parameter [12]. In contrast, under hydrostatic
pressure, the Ir-O-Ir bond angle increases while the lattice parameter
decreases [13]. Hence it would be interesting to explore whether the bandwidth
can be tuned by applying a combination of chemical and hydrostatic pressures
in a systematic manner.

Recently, it has been reported that Sm2Ir2O7 doped with Bi (lattice constant
$a$ = 10.3235 Å for Sm2Ir2O7 and 10.3250 Å for Bi2Ir2O7) shows anomalous
lattice contraction up to 10 % Bi doping, followed by normal lattice expansion
with further Bi substitution [14]. This series of iridates provides an
interesting system to see the effect of the hydrostatic pressure with
systematic chemical substitution. Both Raman spectroscopy and X-ray
diffraction (XRD) are powerful tools for characterizing the structural changes
induced by pressure. Raman spectroscopy enables the investigation of
vibrational modes, providing information about local atomic environments and
structural distortions, while XRD provides information on crystallographic
changes. The synergy between these techniques allows for a comprehensive
analysis of the material’s response to pressure.

In this study, we systematically investigate the (Sm1-xBix)2Ir2O7 (SBIO)
series for $x$ = 0, 0.02, and 0.10 to explore the influence of chemical tuning
on the high-pressure behavior of these compounds. Our goal is to unravel how
the delicate balance between electron-lattice interactions and chemical
composition governs the observed structural transformations. An iso-structural
transition in Sm2Ir2O7 at $\sim$ 11.2 GPa, marked $P_{c}$, is suggested by our
Raman and X-ray data, which is linked to the rearrangement of IrO6 octahedra
in the pyrochlore lattice. The linewidths of the Ir-O-Ir bending ($A_{1g}$ and
$E_{g}$) and Ir-O stretching ($T_{2g}^{4}$) vibrations exhibit anomalous drop
with pressure up to $P_{c}$ for all the three samples. This decrease is caused
by a decrease in the electron-phonon interaction due to the increased
electronic bandwidth of these materials with pressure. The critical pressure
$P_{c}$ decreases to $\sim$ 10.2 and 9 GPa for $x$ = 0.02 and 0.10,
respectively.

##  II Experimental details

All the samples in the SBIO series ($x$ = 0, 0.02, and 0.10) were synthesized
via solid-state reaction method using high purity (99.9 %) precursors of
Sm2O3, IrO2, and Bi2O3. The details of sample synthesis and characterization
are given in [14, 15]. The high-pressure XRD measurements were carried out at
Xpress beamline, Elettra, in Italy, using a monochromatic XRD wavelength
($\lambda$) of 0.4957 Å at room temperature. Standard CeO2 powder sample was
used for calibrating the sample-to-detector distance and orientation angles of
the detector. The SBIO pellets ($x$ = 0, 0.02, and 0.10) were finely ground
and put into membrane-type diamond-anvil cells (DACs) with a culet diameter of
$\sim$ 400 $\mu$m for the XRD measurements. The pressure-transmitting medium
(PTM) used for the membrane-type DAC was methanol: ethanol solution (4:1); and
the applied pressure was measured using a ruby fluorescence marker [16, 17,
18]. Two-dimensional (2D) diffraction patterns were captured using a Dectris
Pilatus 6M detector, and subsequently, these patterns were transformed into
$I$(2$\theta$) diffraction profiles by radial integration of the diffraction
rings using FIT2D software [19]. The structural analysis involved fitting and
refining the raw data using a conventional Rietveld refinement procedure,
facilitated by the GSAS software package [20]. High-pressure Raman scattering
experiments at room temperature were carried out in backscattering geometry
using a 50X objective in a Horiba LabRAM HR Evolution Spectrometer equipped
with a DPSS laser source of wavelength 532 nm and $\sim$ 1.5 mW of laser power
on the sample. The DAC used for Raman measurements was Mao-Bell type with two
16-facet brilliant cut diamonds with culet diameter size of $\sim$ 600 $\mu$m.
The experiments were carried out on a small piece ($\sim$ 75 $\mu$m) taken
from polycrystalline sintered pellets of Bi-doped Sm2Ir2O7 samples. The sample
was placed inside a stainless-steel gasket hole with a diameter of $\sim$ 150
$\mu$m. For doped Sm2Ir2O7, the pressure was applied using a 4:1 methanol-
ethanol combination as PTM that had a freezing pressure of $\sim$ 10.4 GPa
[21]. However, for Sm2Ir2O7, liquid distilled H2O was used as the PTM as the
methanol-ethanol solution was giving rise to a large background and poor
signal-to-noise ratio. The spectra were acquired using a thermoelectrically
cooled charge-coupled device (CCD) (HORIBA Jobin Yvon, SYNCERITY 1024 $\times$
256). The recorded Raman spectra are fitted with a sum of Lorentzian line
shapes using the nonlinear least square fitting method in the Origin software
package [22] to extract phonon frequencies, linewidths, and intensities.

Figure 1:  (a-c) Angle dispersive X-ray diffraction patterns of
(Sm1-xBix)2Ir2O7 (SBIO) series for $x$ = 0, 0.02, and, 0.10, respectively at
selected pressures ranging from 0.2 to 20 GPa. (d-f) Rietveld refined XRD
patterns at 0.5 GPa with the Debye-Scherrer ring pattern shown in the insets.
Solid circles indicate experimental data and the star symbols in the (f) panel
mark the Ir metal impurity peaks. Calculated patterns are drawn as red solid
lines. Bragg reflection positions are indicated by vertical bars. Lower dark
green curves are the difference between the observed and calculated profiles.

##  III Results and Discussions

###  III.1 High-pressure XRD studies

The structural stability of SBIO under pressure was investigated at room
temperature up to $\sim$ 20.4, 19.6, and 20.3 GPa for samples with $x$ = 0,
0.02, and 0.10, respectively. The angle dispersive powder XRD patterns of SBIO
at various high pressures at 300 K are shown in Fig. 1 (a-c). These
diffraction patterns show that all of the Bragg peaks shift in the direction
of increasing 2$\theta$ angles as pressure increases, inferring the expected
gradual contraction of the lattice. Further, the diffraction peaks
significantly broaden at higher pressures due to strain-broadening. However,
the absence of any new Bragg peak or of existing peaks splitting indicates
absence of structural phase transition under pressure. Figure 1 (d-f) shows
the Rietveld fitted diffraction profiles for ambient pressure cubic pyrochlore
phase having space group Fd$\overline{3}$m ($z$ = 8). The lattice parameters,
atomic positions, line shape, and background were all refined in this process
and the goodness-of-fit parameter ($\chi^{2}$) given in Fig. 1 indicates that
the quality of the refinements remains good throughout the SBIO series. In
Fig. 2 (a-c), the volume of the unit cell (estimated using the lattice
parameter value from Rietveld refinement of SBIO series) is plotted against
pressure which shows the cubic lattice volume decreases with increasing
pressure. The $P$-$V$ data are fitted by a third-order Birch-Murnaghan (BM)
equation of state (EOS) [23] given by,

$\displaystyle P=$
$\displaystyle\frac{3B}{2}\left[\left(\frac{V_{0}}{V}\right)^{\frac{7}{3}}-\left(\frac{V_{0}}{V}\right)^{\frac{5}{3}}\right]$
(1)
$\displaystyle\times\left\\{\left[1+\frac{3}{4}\left(B^{\prime}-4\right)\right]\left[\left(\frac{V_{0}}{V}\right)^{\frac{2}{3}}\right]\right\\}$

where $B$ represents the bulk modulus or the modulus of incompressibility at
ambient pressure, $B^{\prime}$ represents the first derivative of the bulk
modulus with respect to pressure, and $V_{0}$ represents the unit cell volume
at ambient pressure. The extracted values from the fit are $B$ = 198.4 $\pm$
4.0 GPa and $B^{\prime}$ = 7.8 $\pm$ 0.8 for $x$ = 0, which compares well with
the previously reported value for Eu2Ir2O7 [24, 25] and recent preprint on
Sm2Ir2O7 [26]. The fit parameters for doped samples are, $B$ = 214.7 $\pm$ 6.3
GPa and $B^{\prime}$ = 8.5 $\pm$ 2.4 for $x$ = 0.02, and $B$ = 250.5 $\pm$ 9.6
GPa and $B^{\prime}$ = 18.5 $\pm$ 4.0 GPa for $x$ = 0.10. The value of
$B^{\prime}$ becomes noticeably large in the case of $x$ = 0.10, indicating
that doped SBIO ($x$ = 0.10) stiffens rapidly under pressure.

Figure 2:  (a-c) Pressure dependence of lattice volume in the pyrochlore phase
of SBIO ($x$ = 0, 0.02, and 0.10, respectively), fitted (solid red line)
$P$-$V$ diagram using a third-order Birch–Murnaghan equation of state. Inset
in the rightmost top panel show the variation of the bulk modulus ($B$) with
doping concentration $x$.(d-o) The pressure dependence of the Rietveld
refinement parameters, Oxygen coordinate ($u$), Ir-O-Ir bond angle
($\theta_{Ir-O-Ir}$), Ir-Ir (dIr-Ir), and Ir-O bond distance (dIr-0),
respectively for SBIO ($x$ = 0, 0.02, and 0.10). The vertical dashed lines in
each panel indicate $P_{c}$ and the linear fit below $P_{c}$ is extrapolated
above $P_{c}$ by dashed red lines in panels m,n, and o.

It is clear from Fig. 2 that the $P$-$V$ plot for all the samples can be
fitted over the entire pressure window upto $\sim$ 20 GPa. However, the
signatures of a transition at $P_{c}$, not reported so far, are clearly seen
in the pressure dependence of the unit cell parameters derived from the
Rietveld refinement plotted in Fig. 2 (d-o). In the pyrochlore structure, each
Ir-site cation is coordinated to six O (48f) ions, forming an IrO6 octahedron
which has a variable $u$-coordinate, the only free parameter in the pyrochlore
structure besides the lattice parameter, $a$. The perfect octahedral symmetry
is achieved for $u$ = 0.3125. The pyrochlore stability field lies within the
interval of 0.3125 $\leq$ $u$ $\leq$ 0.375, and the IrO6 octahedral exhibits a
trigonal compression that increases with $u$ [27, 28, 24]. The evolution of
$u$ as a function of pressure in Fig. 2 (d) shows a finite increase with a
jump near $P_{c}$, suggesting that the IrO6 octahedra become increasingly more
distorted at higher pressures with a sudden change in the degree of distortion
occurring at $P_{c}$, likely due to an iso-structural transition implying a
rearrangement of the IrO6 octahedra. The increase in $u$ in Sm2Ir2O7 with
pressure is similar to Eu2Ir2O7 [24], but in the recent high-pressure preprint
on $A_{2}$Ir2O7 ($A$ = Pr, Sm, Dy-Lu), $u$ remains constant over increasing
pressure [26]. Since the pressure-dependent XRD patterns do not reveal any new
diffraction peak or splitting of lines, it is inferred that the structural
deformation at $\sim$ $P_{c}$ is a local distortion. It is possible that as
the pressure increases, due to vacancies at the 8a sites, the Ir4+ ions adjust
their local coordinates, causing other atoms in the lattice to relocate,
retaining the cubic symmetry and affecting oxygen positions ($u$) in the IrO6
octahedra as reflected in the abrupt change in $u$ near at $P_{c}$ [29]. The
corresponding Ir-O-Ir bond angle (Fig. 2 (g)) and Ir-O bond length (Fig. 2
(j)) also show discontinuities near $P_{c}$. The pressure coefficient of Ir-Ir
bond length (Fig. 2 (m)) also shows a clear change at $P_{c}$. The inset of
Fig. 2 (c) shows the increase of the bulk modulus with the doping
concentration $x$. Increasing values of bulk modulus with increasing $x$ is
expected, following the anomalous lattice contraction in SBIO from $x$ = 0 to
0.10. It may be noted that all the four lattice characteristics for $x$ = 0.02
and 0.10 samples also change at $P$ $\sim$ 10.2 and 9 GPa, respectively in
Fig. 2 (e-f, h-i, k-l, n-o). The decrease in $P_{c}$ with $x$ can be
attributed to the anomalous lattice contraction in SBIO.

Figure 3: (a) Fitted Raman spectra of Sm2Ir2O7 at some representative
pressures. Dark green curves are individual fit to the phonon modes, and solid
red curves are cumulative fit to the spectra. (b-c) The pressure dependence of
frequencies and linewidths, respectively, of various phonon modes up to 25
GPa. Solid red and blue lines are linear fit to the data below and above
$P_{c}$, respectively. The vertical dashed lines indicate $P_{c}$ and the
linear fit below $P_{c}$ is extrapolated above $P_{c}$ by dashed red lines in
(b) and (c).

###  III.2 High-pressure Raman studies on Sm2Ir2O7

Based on group theory, the pyrochlore structure (space group
Fd$\overline{3}$m) has zone-center optical, $\Gamma_{op}$ = $A_{1g}$ \+
$E_{g}$ \+ 4$T_{2g}$ \+ 7$T_{1u}$ and acoustic, $\Gamma_{ac}$ = $T_{1u}$
phonon modes [30]. The Raman active phonons are $A_{1g}$, $E_{g}$, and
4$T_{2g}$, and the rest seven are IR active. The Raman modes are assigned with
irreducible representations, as shown in Fig. 3 (a), based on prior reports on
pyrochlore iridates [30, 31]. The mode assignments are as follows : $A_{1g}$
and $E_{g}$ modes involve Ir-O-Ir bending, $T_{2g}^{1}$ and $T_{2g}^{2}$ modes
associated with Sm-O stretching, $T_{2g}^{3}$ mode due to the vibration of O’
ions surrounded by eight Sm ions and $T_{2g}^{4}$ mode related to Ir-O
stretching vibration [30]. As the cation sites of Ir4+ and Sm3+ have center of
inversion, the Raman active modes involve only the vibrations of oxygen atoms.
Figure S1 shows the pressure evolution of the Raman modes at some
representative pressures. Raman spectrum at the lowest pressure inside the DAC
for Sm2Ir2O7 is plotted in the bottommost panel of Fig. 3 (a), which shows
five phonon modes. The position and relative intensities of these bands agree
well with the literature [30, 31]. $T_{2g}^{3}$ is a weak mode and could not
be detected inside the DAC. Raman spectra at an intermediate and highest
pressure are shown in the middle and topmost panel in Fig. 3 (a). The same
number of phonon modes are observed with varying pressure which confirms the
preservation of crystal symmetry with pressure, in agreement with the XRD
results. The individual phonon modes were fitted with Lorentzian line shape as
shown in Fig. 3 (a), and the extracted phonon frequencies and linewidths are
plotted against pressure in Figs. 3 (b) and (c). The straight lines represent
the fitting of mode frequencies using the linear equation,
$\omega_{P}=\omega_{0}+\left(\frac{d\omega}{dP}\right)P$, where $\omega_{0}$
is the mode frequency at ambient pressure. All the phonon modes demonstrate
expected hardening with increasing pressure due to the compression of the unit
cell, resulting in an increase in the stiffness constant ($K$) associated with
the vibrational mode. However, we observe an alteration of the pressure-
coefficients ($d\omega/dP$) of the phonon mode frequencies across $P_{c}$
$\sim$ 11.2 GPa, supporting the presence of an iso-structural transition as
inferred by the XRD results. It should be noted that we have performed the
high-pressure experiment with distilled water as the PTM, which starts
freezing above 1 GPa, making it a quasi-hydrostatic pressure medium.
Therefore, the anomalies near $P_{c}$ are unlikely to be triggered by the non-
hydrostatic nature of the medium. Further, our XRD data using methanol:
ethanol as PTM also exhibit anomalies near $P_{c}$, indicating that the
transition is intrinsic to the system. A previous high-pressure study on
Gd2Ti2O7 has also shown that the pressure dependence of the Raman modes is
seen to be similar in both methanol-ethanol mixture and distilled water as PTM
[32]. The two high-pressure iso-structural phases are associated with a
positive slope value for all the phonon modes, as depicted in Fig. 3 (b). To
quantify the phonon frequency changes under pressure we have estimated the
Grüneisen parameter ($\gamma$) given by,

$\gamma=\frac{B}{\omega_{0}}\frac{\mathrm{d}\omega}{\mathrm{d}P}$ (2)

below and above $P_{c}$ for all the phonon modes. The values are listed in
Table I. The significant changes in $\gamma$ suggest a subtle structural
deformation, as seen in pressure-dependent XRD measurements (see Fig. 2).

Modes | $P<P_{c}$ | $P>P_{c}$
---|---|---
$T_{2g}^{1}$ | 1.04 | 1.35
$E_{g}$ | 1.28 | 1.42
$T_{2g}^{2}$ | 1.35 | 0.95
$A_{1g}$ | 1.12 | 0.72
$T_{2g}^{4}$ | 1.25 | 0.85
Table 1:  Grüneisen parameter ($\gamma$) for various phonon modes below and
above $P_{c}$ for $x$ = 0

To understand the frequency variations in Sm2Ir2O7, we compared the effect of
physical and chemical pressure, similar to our earlier high-pressure work on
Eu2Ir2O7 [30]. As an effect of $A$\- site cation radius, the lattice volume
decreases by $\sim$ 32 ${\AA}^{3}$, coming from Pr2Ir2O7 to Sm2Ir2O7 in the
pyrochlore series [33]. The difference in phonon frequencies for the $A_{1g}$
mode in Sm2Ir2O7 compared to Pr2Ir2O7 is $\sim$ 2 cm-1 [30]. Our $P$-$V$ data
(Fig. 2 (a)) indicates that a volume reduction of 32 ${\AA}^{3}$ equates to a
pressure of $\sim$ 7 GPa, causing the $A_{1g}$ mode frequency to harden by
$\sim$ 18 cm-1. The significant difference in frequency shifts for the
$A_{1g}$ phonon due to physical and chemical pressures cannot be explained
only by the quasiharmonic effect, indicating that the $A_{1g}$ phonon is also
renormalized by an underlying electron-phonon interaction. It can be seen from
Fig. 2 that the Ir-Ir and Ir-O bond distances systematically decrease up to
$P_{c}$, and thus increasing electronic bandwidth in a metal by enhancing the
hybridization of the Ir-5$d$ and O-2$p$ orbitals, and direct hopping between
the Ir sites [12, 13]. Similarly, an increase in the Ir-O-Ir bond angle would
favor electrons hopping between two Ir atoms via the oxygen atom and
contribute to the increasing bandwidth. Thus, the large frequency variation of
the $A_{1g}$ mode in Sm2Ir2O7 can be correlated with the increase in the
electronic bandwidth leading to significant changes in the electron-phonon
interaction [34].

Figure 3 (c) depicts the pressure evolution of linewidths for the five
strongest Raman phonons, showing an unexpected drop in the linewidth of the
Ir-O-Ir bending ($A_{1g}$ and $E_{g}$) as well as Ir-O stretching vibrations
$T_{2g}^{4}$ upto $P_{c}$. The solid lines are linear fit to
$\Gamma=\Gamma_{0}+\left(\frac{d\Gamma}{dP}\right)P$. It can be seen that
$\frac{d\Gamma}{dP}$ is negative for $A_{1g}$, $E_{g}$, and $T_{2g}^{4}$
modes. The values of $\frac{d\Gamma}{dP}$ are given in table II. This
observation is similar to our recent high-pressure Raman study on Eu2Ir2O7,
where an equivalent linewidth drop was seen for the $A_{1g}$ phonon [25]. One
possible explanation for the anomalous decrease in the linewidth of these
modes is a decrease in the electron-phonon interaction due to pressure. Since
this anomaly is observed in phonons related to Ir-O and Ir-O-Ir vibrations, it
can be correlated with the observed systematic trend of the Ir–Ir and Ir-O
bond distance as well as the Ir-O-Ir bond angle up to $P_{c}$ which increases
the electronic bandwidth, thereby decreasing the electron-phonon interaction.
According to the Holstein model, the electronic-phonon interaction strength in
a metal, $\lambda=2g^{2}/W\omega$ (where $\omega$ is the phonon frequency, $t$
is the nearest-neighbor hopping integral, $W$ = 8$t$ is the electronic
bandwidth, and $g$ is the coupling constant), will vary inversely with its
electronic bandwidth [35]. As seen in Fig. 2, $\theta_{Ir-O-Ir}$ increases
with pressure, increasing the electronic bandwidth and thus decreasing
$\lambda$ and the observed linewidth drop. The linewidth of other $T_{2g}^{1}$
and $T_{2g}^{2}$ phonons increases with increasing pressure which can be
attributed to the corresponding change in the two-phonon density of states,
which affects the anharmonic interactions [36].

Figure 4:  (a,d) Fitted Raman spectra of SBIO ($x$ = 0.02 and 0.10, respectively) at some representative pressures. ((b,c) and (e,f) The pressure dependence of frequencies and linewidths of various phonon modes for $x$ = 0.02 and 0.10, respectively. Symbols and lines have the same meaning as Fig. 3. | $x=0$ | $x=0.02$ | $x=0.10$
---|---|---|---
Modes | $P<P_{c}$ | $P>P_{c}$ | $P<P_{c}$ | $P>P_{c}$ | $P<P_{c}$ | $P>P_{c}$
$T_{2g}^{1}$ | 1.4 $\pm$ 0.3 | 1.5 $\pm$ 0.2 | 0.45 $\pm$ 0.2 | 1.5 $\pm$ 0.4 | 0.31 $\pm$ 0.3 | -
$E_{g}$ | -0.2 $\pm$ 0.08 | 0 | -0.5 $\pm$ 0.2 | 0.02 $\pm$ 0.07 | -1 $\pm$ 0.04 | 0
$T_{2g}^{2}$ | 0.13 $\pm$ 0.05 | 0.62 $\pm$ 0.05 | 0.3 $\pm$ 0.02 | 0.3 $\pm$ 0.02 | 0.44 $\pm$ 0.02 | 0.44 $\pm$ 0.02
$A_{1g}$ | -0.4 $\pm$ 0.04 | 0.25 $\pm$ 0.03 | -0.4 $\pm$ 0.03 | 0.14 $\pm$ 0.03 | -1.2 $\pm$ 0.3 | 0
$T_{2g}^{4}$ | -1.6 $\pm$ 0.3 | 1.1 $\pm$ 0.17 | -0.7 $\pm$ 0.1 | 0.43 $\pm$ 0.1 | -0.75 $\pm$ 0.2 | 0.87 $\pm$ 0.14
Table 2:  $\frac{d\Gamma}{dP}$ for various phonon modes below and above
$P_{c}$

###  III.3 High-pressure Raman studies on SBIO ($x$ = 0.02 and 0.10)

The pressure evolution of the Raman spectra of $x$ = 0.02 and 0.10 are given
in Figs. S2-S3, showing the same number of phonon modes up to the highest
measured pressure. Figure 4 (a) and (d) shows fitted Raman spectra for SBIO
($x$ = 0.02 and 0.10, respectively) at a few representative pressures, marking
$E_{g}$, $A_{1g}$, and three $T_{2g}$ phonons. The corresponding phonon
frequencies are presented in panels (b) and (e) of this figure, with pressure
coefficients listed alongside the straight-line fits to the data. For all
phonons, the frequency versus pressure plot fits with a single positive slope
($\frac{d\omega}{dP}$ ) over the whole recorded pressure range. The lack of
slope change at $P_{c}$ in doped Sm2Ir2O7 can be understood by the fact that
the IrO6 octahedra are less deformed at ambient pressure due to a smaller
value of the oxygen coordinate $u$, and the iso-structural rearrangement is
smaller as compared to the undoped sample. The linewidth of the $A_{1g}$,
$E_{g}$, and $T_{2g}^{4}$ modes show anomalous decrease up to $P_{c}$ ($\sim$
10.2 and 9 GPa for x = 0.02 and 0.10, respectively) as shown in panels (c) and
(f) of Fig. 4, similar to the $x$ = 0 sample. Figure 5 (a) depicts the
evolution of the pressure coefficient of the linewidth ($\frac{d\Gamma}{dP}$)
as a function of doping concentration ($x$) which demonstrates that the
$T_{2g}^{4}$ phonon for $x$ = 0 has a greater linewidth decrease than the
$A_{1g}$ and $E_{g}$ phonons, whose amplitude decreases with Bi doping. The
$\frac{d\Gamma}{dP}$ of the $A_{1g}$ and $E_{g}$ phonons is similar for $x$ =
0 and 0.02 but increases significantly for $x$ = 0.10 sample. These variations
in the linewidths of $A_{1g}$, $E_{g}$, and $T_{2g}^{4}$ modes infer that the
Ir-O-Ir bending and Ir-O bond stretching vibrations are influenced differently
by electron-phonon interaction. Our Raman data clearly reveal that the
crossover pressure $P_{c}$ for the doped sample decreases gradually with $x$
similar to (Eu1-xBix)2Ir2O7 [25].

Figure 5:  (a,b) Evolution of $\frac{d\Gamma}{dP}$ and Grüneisen parameter
below $P_{c}$ ($\gamma$ ), respectively of various Raman phonons with doping
concentration $x$. Dashed lines are guides to the eye.

Figure 5 (b) shows the evolution of Grüneisen parameter ($\gamma$) (below
$P_{c}$) with doping concentration $x$ for five phonon modes, showing a
monotonic increase, as in (Eu1-xBix)2Ir2O7 [25], reflecting the increase of
bulk modulus with $x$ (see inset of Fig. 2 (c)).

##  IV Conclusions

In summary, we have investigated pressure-induced structural changes in the
(Sm1-xBix)2Ir2O7 ($x$ = 0, 0.02, and 0.10) series using synchrotron-based
X-ray diffraction and Raman-scattering measurements up to $\sim$ 20 and 25
GPa, respectively. Our Raman and X-ray results suggest an iso-structural phase
transition associated with the rearrangement of IrO6 octahedra in the
pyrochlore lattice in Sm2Ir2O7, at $\sim$ 11.2 GPa. The transition pressure
decreases to $\sim$ 10.2 and 9 GPa for $x$ = 0.02 and 0.10, respectively due
to an anomalous lattice contaction in SBIO series. The linewidths of the Ir-O-
Ir bending ($A_{1g}$ and $E_{g}$) and Ir-O stretching ($T_{2g}^{4}$)
vibrations exhibit anomalous drop with increasing pressure up to $P_{c}$ for
all the samples, attributed to a decrease in electron-phonon interaction. The
integration of Raman spectroscopy and XRD measurements in this study not only
enhances the reliability of our observations but also allows for a
comprehensive understanding of the structural changes occurring in SBIO under
high pressure. Our findings contribute to the fundamental understanding of
pyrochlore iridates and hold potential implications for the design of
materials with tailored properties and novel quantum states.

##  V Acknowledgement

AKS thanks the Department of Science and Technology, Government of India, for
financial support under the National Science Chair Professorship. The authors
acknowledge the financial support by the Department of Science and Technology
(DST) of the Government of India for the high-pressure XRD measurements at the
Xpress beamline of Elettra Sincrotrone, Trieste.

Figure 6:  [Fig.S1 : Pressure evolution of the Raman spectra for $x$ = 0.]
Figure 7: [Fig.S2 : Pressure evolution of the Raman spectra for $x$ = 0.02.]
Figure 8: [Fig.S3 : Pressure evolution of the Raman spectra for $x$ = 0.10.]

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