# Hypergraph Turán Problems in $\ell_{2}$-Norm

József Balogh 111Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA, and Moscow Institute of Physics and
Technology, Russian Federation. E-mail<EMAIL_ADDRESS>Research is
partially supported by NSF Grant DMS-1764123, NSF RTG grant DMS 1937241,
Arnold O. Beckman Research Award (UIUC Campus Research Board RB 18132), the
Langan Scholar Fund (UIUC), and the Simons Fellowship. Felix Christian Clemen
222Department of Mathematics, University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA, E-mail<EMAIL_ADDRESS>Research of this
author is partially supported by the Arnold O. Beckman Research Award (UIUC
Campus Research Board RB 18132). Bernard Lidický 333Iowa State University,
Department of Mathematics, Iowa State University, Ames, IA, E-mail:
<EMAIL_ADDRESS>Research of this author is partially supported by NSF
grant DMS-1855653.

(September 3, 2024)

###### Abstract

There are various different notions measuring extremality of hypergraphs. In
this survey we compare the recently introduced notion of the codegree squared
extremal function with the Turán function, the minimum codegree threshold and
the uniform Turán density.

The codegree squared sum $\textup{co}_{2}(G)$ of a $3$-uniform hypergraph $G$
is defined to be the sum of codegrees squared $d(x,y)^{2}$ over all pairs of
vertices $x,y$. In other words, this is the square of the $\ell_{2}$-norm of
the codegree vector. We are interested in how large $\textup{co}_{2}(G)$ can
be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This
maximum value of $\textup{co}_{2}(G)$ over all $H$-free $n$-vertex $3$-uniform
hypergraphs $G$ is called the codegree squared extremal function, which we
denote by $\textup{exco}_{2}(n,H)$.

We systemically study the extremal codegree squared sum of various $3$-uniform
hypergraphs using various proof techniques. Some of our proofs rely on the
flag algebra method while others use more classical tools such as the
stability method. In particular, we (asymptotically) determine the codegree
squared extremal numbers of matchings, stars, paths, cycles, and $F_{5}$, the
$5$-vertex hypergraph with edge set $\\{123,124,345\\}$.

Additionally, our paper has a survey format, as we state several conjectures
and give an overview of Turán densities, minimum codegree thresholds and
codegree squared extremal numbers of popular hypergraphs.

##  1 Introduction

Given a $k$-uniform hypergraph (or $k$-graph) $H$, the _Turán function_ (or
_extremal number_) $\textup{ex}(n,H)$ is the maximum number of edges in an
$H$-free $n$-vertex $k$-uniform hypergraph. The _Turán density_ of $H$, denote
by $\pi(H)$, is the scaled limit

$\pi(H)=\lim_{n\to\infty}\frac{\textup{ex}(n,H)}{\binom{n}{k}}.$

Determining these numbers is a central problem in extremal combinatorics. For
graphs ($k=2$), this question is well-explored. The Erdős-Stone theorem [23,
22] asymptotically determines the Turán density for graphs with chromatic
number at least three. For hypergraphs, determining the Turán density is
notoriously difficult; very few exact results are known. For example, the
Turán density of the innocent looking _tetrahedron_ $K_{4}^{3}$, the complete
$3$-uniform hypergraph on $4$ vertices, is unknown.

In order to get a better understanding of these problems, various different
kinds of extremality such as the generalized Turán function or the minimum
codegree threshold have been studied. We [3] recently introduced a new type of
extremality for hypergraphs and solved the tetrahedron problem asymptotically
for this notion. Here, we will systematically study extremal problems
regarding this function.
Let $G$ be an $n$-vertex $k$-uniform hypergraph. For a vertex set $T\subset
V(G)$, the _codegree_ of $T$, denoted by $d_{G}(T)$, is the number of edges in
$G$ containing $T$. We drop the index if $G$ is clear from the context. The
_codegree vector_ of $G$ is the vector $X\in\mathbb{Z}^{\binom{V(G)}{k-1}}$,
where $X(v_{1},v_{2},\ldots,v_{k-1})=d_{G}(v_{1},v_{2},\ldots,v_{k-1})$ for
all $\\{v_{1},v_{2},\ldots,v_{k-1}\\}\in\binom{V(G)}{k-1}$. Finding
$\textup{ex}(n,H)$ is equivalent to determining the maximum $\ell_{1}$-norm of
the codegree vector of an $H$-free $n$-vertex $k$-uniform hypergraph. Here, we
study maximality with respect to the $\ell_{2}$-norm of the codegree vector.
The _codegree squared sum_ $\textup{co}_{2}(G)$ is the sum of codegrees
squared over all $k-1$ sets $T$, i.e.,

$\displaystyle\textup{co}_{2}(G)=\sum_{\begin{subarray}{c}T\in\binom{[n]}{k-1}\end{subarray}}d_{G}^{2}(T).$

In other words, the codegree squared sum is the square of the $\ell_{2}$-norm
of the codegree vector.

######  Question 1.1.

[3] Given a $k$-uniform hypergraph $H$, what is the maximum $\ell_{2}$-norm of
the codegree vector of a $k$-uniform $H$-free $n$-vertex hypergraph $G$?

We follow the notation introduced in [3]. Let $\mathcal{F}$ be a family of
$k$-uniform hypergraphs. Denote by $\textup{exco}_{2}(n,\mathcal{F})$ the
maximum codegree squared sum among all $k$-uniform $n$-vertex
$\mathcal{F}$-free hypergraphs, and let the codegree squared density
$\sigma(\mathcal{F})$ be its scaled limit, i.e.,

$\displaystyle\textup{exco}_{2}(n,\mathcal{F})=\smash{\displaystyle\max_{G\text{
is }\mathcal{F}\text{-free}}}\textup{co}_{2}(G)\quad\quad\text{ and
}\quad\quad\sigma(\mathcal{F})$
$\displaystyle=\limsup\limits_{n\rightarrow\infty}\frac{\textup{exco}_{2}(n,\mathcal{F})}{\binom{n}{k-1}(n-k+1)^{2}}.$
(1)

We [3] proved general properties of $\sigma$ including the existency of the
limit in (1). In Table 1 we present bounds and exact values for the codegree
squared density of various hypergraphs. Table LABEL:tab:pic provides the
definitions with pictures of all hypergraphs included in Table 1. Unless
otherwise mentioned, all upper bounds on $\sigma$ in this table were obtained
using Razborov’s flag algebra machinery [57]. It is a standard application of
flag algebras to obtain these results; we give a short explanation of it in
Section 2.3. Table 1 also gives an overview of known results for the Turán
density, the minimum codegree threshold and the uniform Turán density. Denote
by $\delta(G)$ the minimum $(k-1)$-codegree of a $k$-graph $G$. For a family
of $k$-graphs $\mathcal{F}$, the _minimum codegree Turán number_
$\textup{ex}_{k-1}(n,\mathcal{F})$ is the maximum $\delta(G)$ over all
$\mathcal{F}$-free $k$-graphs $G$ on $n$ vertices. The _minimum codegree
threshold_ ,

$\pi_{k-1}(\mathcal{F})=\lim_{n\to\infty}\frac{\textup{ex}_{k-1}(n,\mathcal{F})}{\binom{n}{k-2}},$

is its scaled limit.
Reiher, Rödl and Schacht [60] recently introduced a variant of the Turán
density, where we want to maximize the density of every linear sized subsets
of $H$-free hypergraphs. For real numbers $d\in[0,1]$, and $\eta>0$ a
$3$-graph $G=(V,E)$ is _$(d,\eta,1)$ -dense_ if for all $U\subseteq V$ the
relation

$\left|U^{(3)}\cap E\right|\geq d\binom{|U|}{3}-\eta|V|^{3}$

holds, where $U^{(3)}$ denotes the set of all three element subsets of $U$.
The _uniform Turán density_ $\pi_{u}(H)$ of a 3-graph $H$ is defined to be

$\displaystyle\pi_{u}(H):$ $\displaystyle=\sup\\{d\in[0,1]:\ \text{ for every
}\eta>0\text{ and }n\in\mathbb{N}\text{ there exists an }\text{$H$-free}$
$\displaystyle\quad\ \ \text{$(d,\eta,1)$-dense hypergraph }G\text{ with
}|V(G)|\geq n\\}.$

######  Theorem 1.2.

All bounds presented in Table 1 hold.

Here we collect many known results, and when known results are lacking, we at
least mention the ‘trivial’ bounds. Most upper bounds were obtained by a
simple application of flag algebras.

In addition to the bounds presented in Table 1, we asymptotically determine
the maximum $\ell_{2}$-norm of $3$-graphs not containing a loose cycle, loose
path, matching or star. These problems are not approachable with flag algebra
methods due to the fact that their codegree squared extremal number is
$o(n^{4})$. We use non-computer assisted methods to obtain these results. The
discussion about the history of these problems will be deferred to the
corresponding sections. Additionally, we provide a non-computer assisted proof
determining the exact codegree squared extremal number of $F_{5}$.

Denote by $S_{n}$ the complete $3$-partite $3$-graph on $n$ vertices with part
sizes $\lfloor n/3\rfloor,\lfloor(n+1)/3\rfloor,$ $\lfloor(n+2)/3\rfloor$. We
[3] showed that $S_{n}$ is the largest $\\{F_{4},F_{5}\\}$-free $3$-graph in
$\ell_{2}$-norm using a simple double counting argument and the corresponding
$\ell_{1}$-norm result by Bollobás [5]. Here, we will expand this result for
$F_{5}$-free $3$-graphs, which requires more work than just applying the
corresponding $\ell_{1}$-norm result.

$H$ | $\pi(H)\geq$ | $\pi(H)\leq$ | $\sigma(H)\geq$ | $\sigma(H)\leq$ | $\pi_{2}(H)\geq$ | $\pi_{2}(H)\leq$ | $\pi_{u}(H)\geq$ | $\pi_{u}(H)\leq$
---|---|---|---|---|---|---|---|---
$K_{4}^{3}$ |  $5/9$[67] |  $0.5615$[1] |  $1/3$[3] |  $1/3$[3] |  $1/2$ [56] | 0.529 | $1/2$ | 0.529
$K_{5}^{3}$ |  $3/4$[62, 67] |  $0.7696$ [68] |  $5/8$[3] |  $5/8$[3] |  $2/3$ [50, 24] | $0.74$ | $2/3$ | 0.758
$K_{6}^{3}$ |  $0.84$[62, 67] |  $0.8584$[68] |  $0.7348$[3] | $0.7536$ |  $3/4$ [50, 24] | $0.838$ | $3/4$ | 0.853
$F_{3,2}$ |  $4/9$ [54] |  $4/9$ [35] | $1/4$ | $1/4+10^{-9}$ |  $1/3$ [25] |  $1/3$ [25] | $0$ |  $0$ [60]
$F_{3,3}$ |  $3/4$[54] |  $3/4$[54] |  $5/8$[3] |  $5/8$[3] | $1/2$ | $0.604$ | $1/4$ | 0.605
$F_{5}$ |  $2/9$ [5] |  $2/9$ [30] | $2/27$ | $2/27$ | $0$ | $0$ | $0$ |  $0$ [60]
$\mathbb{F}$ |  $3/4$ [65] |  $3/4$ [16] | $5/8$ | $3/4$ |  $1/2$[53] |  $1/2$ [53] | 0 |  0 [60]
$K_{4}^{3-}$ |  $2/7$ [31] |  $0.28689$ [68] | $4/43$ | $0.09307$ |  $1/4$ [56] |  $1/4$[26] |  $1/4$ [39] |  $1/4$ [39]
$K_{4}^{3-},F_{3,2},C_{5}$ |  $12/49$ [27] |  $12/49$ [27] | $2/27$ | $2/27$ | $1/12$ | $0.186$ | $0$ |  $0$ [60]
$K_{4}^{3-},F_{3,2}$ |  $5/18$ [27] |  $5/18$ [27] | $5/54$ | $5/54$ | $1/12$ | 0.202 | $0$ |  $0$ [60]
$K_{4}^{3-},C_{5}$ |  $1/4$ [27] |  $0.25108$ [27] | $1/13$ | $0.07695$ | $1/12$ | $0.204$ | 0 | 0.251
$F_{3,2},J_{4}$ |  $3/8$ [27] |  $3/8$ [27] | $3/16$ | $3/16$ | $1/12$ | $0.274$ | $0$ |  $0$ [60]
$F_{3,2},J_{5}$ |  $3/8$ [27] |  $3/8$ [27] | $3/16$ | $3/16$ | $1/12$ | $0.28$ | $0$ |  $0$ [60]
$J_{4}$ |  $1/2$ [6] |  $0.50409$ [68] | $0.28$ | $0.2808$ | $1/4$ | $0.473$ | $1/4$ | 0.49
$J_{5}$ |  $1/2$ [6] | $0.64475$ | $0.28$ | $0.44275$ | $1/4$ | 0.613 | $1/4$ | 0.631
$C_{5}$ |  $2\sqrt{3}-3$ [54] |  $0.46829$ [68] | $0.25194$ | $0.25311$ | $1/3$ | $0.3993$ | $1/27$ | 0.402
$C_{5}^{-}$ |  $1/4$ [54] |  $0.25074$ [68] | $1/13$ | $0.07726$ | $0$ | $0.136$ | $0$ |  $0$ [60]
$K_{5}^{=}$ | $0.58656$ | $0.60962$ | $0.35794$ | $0.38873$ | $1/2$ | $0.569$ | $1/2$ | 0.567
$K_{5}^{<}$ | $5/9$ | $0.57705$ | $1/3$ | $0.34022$ | $1/2$ | $0.560$ | $1/2$ | 0.568
$K_{5}^{-}$ | $0.58656$ | $0.64209$ | $0.35794$ | $0.41962$ | $1/2$ | 0.621 | $1/2$ | 0.626
Table 1:  This table shows upper and lower bounds on $\sigma$ and displays the known results for $\pi,\pi_{2}$ and $\pi_{u}$ for various hypergraphs. For the definition of the hypergraphs see Table LABEL:tab:3uniformfig. Useful approximations: $2/7\approx 0.285714$, $2\sqrt{3}-3\approx 0.4641016$, $1/13\approx 0.0769230$, $4/43\approx 0.0930232$, $1/12\approx 0.0833333$.  Table 2: Description of some $3$-uniform hypergraphs.  $H$ | Edges | Visual 1 | Visual 2
---|---|---|---
$K_{4}^{3}$ | 123, 124, 134, 234 |  |
$K_{4}^{3-}=F_{4}$ | 123, 124, 134 |  |
$F_{3,2}$ | 123, 145, 245, 345 |  |
$J_{4}$ | 123, 124, 125, 134, 135, 145 |  |
$J_{5}$ | 123, 124, 125, 126, 134, 135, 136, 145, 146, 156 |  |
$F_{3,3}$ | 123, 145, 146, 156, 245, 246, 256, 345, 346, 356 |  |
$F_{5}$ | 123, 124, 345 |  |
$\mathbb{F}$ | 123, 345, 156, 246, 147, 257, 367 |  |
$C_{5}$ | 123, 234, 345, 145, 125 |  |
$C_{5}^{-}$ | 123, 234, 345, 145 |  |
$\overline{K_{5}^{-}}$ | 123 |  |
$\overline{K_{5}^{=}}$ | 123, 345 |  |
$\overline{K_{5}^{<}}$ | 123, 234 |  |

######  Theorem 1.3.

There exists a number $n_{0}$ such that for all $n\geq n_{0}$

$\displaystyle\textup{exco}_{2}(n,F_{5})=\textup{co}_{2}(S_{n}).$

Furthermore, $S_{n}$ is the unique $F_{5}$-free $3$-graph $G$ on $n$ vertices
satisfying $\textup{co}_{2}(G)=\textup{exco}_{2}(n,F_{5})$.

The _loose $s$-path_ $P^{k}_{s}$ is the $k$-graph with $s$ edges
$\\{e_{1},\ldots,e_{s}\\}$ such that $|e_{i}\cap e_{j}|=1$ if $|i-j|=1$ and
$e_{i}\cap e_{j}=\emptyset$ otherwise. The _loose $s$-cycle_ $C^{k}_{s}$ is
the $k$-uniform hypergraph with $s$ edges $\\{e_{1},\ldots,e_{s}\\}$ obtained
from an $(s-1)$-path $\\{e_{1},\ldots,e_{s-1}\\}$ by adding an edge $e_{s}$
that shares one vertex with $e_{1}$, another vertex with $e_{s-1}$ and is
disjoint from the other edges.

######  Theorem 1.4.

Let $s\geq 4$. Then,

$\displaystyle\textup{exco}_{2}(n,C^{3}_{s})=\left\lfloor\frac{s-1}{2}\right\rfloor
n^{3}(1+o(1))\quad\quad\text{and}\quad\quad\textup{exco}_{2}(n,P^{3}_{s})=\left\lfloor\frac{s-1}{2}\right\rfloor
n^{3}(1+o(1)).$

Denote by $M_{s}^{3}$ the _$3$ -uniform matching_ of size $s$, i.e., the
$3$-uniform hypergraph on $3s$ vertices with $s$ pairwise disjoint edges.

######  Theorem 1.5.

Let $s\geq 2$. Then,

$\displaystyle\textup{exco}_{2}(n,M_{s}^{3})=(s-1)n^{3}(1+o(1)).$

Denote by $S_{s}^{3}$ the _star_ with $s$ edges such that the intersection of
any pair of edges is exactly the same vertex.

######  Theorem 1.6.

Let $s\geq 3$. If $s$ is odd, then

$\displaystyle\textup{exco}_{2}(n,S_{s}^{3})=s(s-1)n^{2}(1+o(1)).$

If $s$ is even, then

$\displaystyle\textup{exco}_{2}(n,S_{s}^{3})=\left(s^{2}-\frac{3}{2}s\right)n^{2}(1+o(1)).$

In this work, we are not intending to duplicate or replace the excellent
survey [44] by Keevash; our aim is to supplement it with new results and
directions.

Our paper is organized as follows; in Section 2 we explain our notation and
explain the flag algebra technique briefly. In Section 3 we present the
constructions leading to the bounds on $\sigma,\pi,\pi_{u},\pi_{2}$ in Table 1
and state conjectures on some of the non-sharp results. In Section 4 we
present the proof of Theorem 1.3. In Sections 5, 6 and 7 we prove Theorems
1.4, 1.5 and 1.6, respectively. Finally, we present related questions for
higher uniformities in Section 8.

##  2 Preliminaries

###  2.1 Terminology and notation

Let $H$ be a $3$-uniform hypergraph, $x,y,z\in V(H)$ and $A,B,C\subseteq V(H)$
be pairwise disjoint sets. We will use the following notation throughout the
paper.

  * •

For an edge $e=\\{x,y,z\\}\in E(H)$ we write $xyz$ for convenience.

  * •

Denote by $L(x)$ the link graph of $x$, i.e., the graph on
$V(H)\setminus\\{x\\}$ with $ab\in E(L(x))$ iff $abx\in E(H)$.

  * •

Denote by $L_{A}(x)=L(x)[A]$ the induced link graph on $A$.

  * •

Denote by $L_{A,B}(x)$ the subgraph of the link graph of $x$ only containing
edges between $A$ and $B$, i.e., $V(L_{A,B}(x))=V(H)\setminus\\{x\\}$ and
$ab\in E(L_{A,B}(x))$ iff $a\in A,b\in B$ and $abx\in E(H)$.

  * •

Denote by $L^{c}_{A,B}(x)$ the subgraph of the link graph of $x$ only
containing non-edges between $A$ and $B$, i.e.,
$V(L_{A,B}(x))=V(H)\setminus\\{x\\}$ and $ab\in E(L^{c}_{A,B}(x))$ iff $a\in
A,b\in B$ and $abx\not\in E(H)$.

  * •

$e(A,B)$ denotes the number of cross-edges between $A$ and $B$, this means
$e(A,B):=|\\{xyz\in E(H):x,y\in A,z\in B\\}|+|\\{xyz\in E(H):x,y\in B,z\in
A\\}|.$

  * •

Denote $e(A,B,C)$ the number of cross-edges between $A,B$ and $C$, i.e.,
$e(A,B,C):=|\\{abc\in E(H):a\in A,b\in B,c\in C\\}|$.

  * •

Let $e=xyz\in E(H)$ be an edge. Define the weight of $e$ to be

$w_{H}(e)=d(x,y)+d(x,z)+d(y,z).$

We drop the index if the hypergraph $H$ is clear from the context.

  * •

The shadow graph $\delta H$ is the subset of all two element subsets of $V(H)$
that contains all pairs contained in some $e\in E(H)$.

######  Definition 2.1.

Let $H$ be a $k$-graph and $t\in\mathbb{N}$. The _blow-up_ $H(t)$ of $H$ is
the $k$-graph obtained by replacing each vertex $x\in V(H)$ by $t$ vertices
$x^{1},\ldots,x^{t}$ and each edge $x_{1}\cdots x_{k}\in E(H)$ by $t^{k}$
edges $x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}$ with $1\leq a_{1},\ldots,a_{k}\leq
t$.

###  2.2 The uniform Turán density

Recently, Reiher, Rödl and Schacht [60] introduced the uniform Turán density.
They characterised $3$-graphs $H$ satisfying $\pi_{u}(H)=0$.

######  Theorem 2.2 (Reiher, Rödl and Schacht [60]).

For a $3$-uniform hypergraph $H$, the following are equivalent:

  * (a)

$\pi_{u}(H)=0$.

  * (b)

There is a labeling of the vertex set $V(H)=\\{v_{1},\ldots,v_{f}\\}$ and
there is a three-colouring $\phi:\delta F\rightarrow\\{\text{red, blue,
green}\\}$ of the pairs of vertices covered by hyperedges of $H$ such that
every hyperedge $v_{i}v_{j}v_{k}\in E(H)$ with $i<j<k$ satisfies

$\displaystyle\phi(v_{i},v_{j})=\text{red},\quad\phi(v_{i},v_{k})=\text{blue},\quad\phi(v_{j},v_{k})=\text{green}.$

We will use their result to observe that $\pi_{u}(H)=0$ for $H=F_{3,2},F_{5}$
and $\mathbb{F}$ in Section 3. Reiher, Rödl and Schacht [60] also observed
that the uniform Turán density has a jump from $0$ to $1/27$.

######  Corollary 2.3 (Reiher, Rödl and Schacht [60]).

If a $3$-graph $H$ satisfies $\pi_{u}(H)>0$, then
$\pi_{u}(H)\geq\frac{1}{27}$.

Somewhat surprisingly, it was not easy to prove that $1/27$ is best possible,
which was done by Garbe, Král and Lamaison [37].

###  2.3 Flag Algebras

We use flag algebras to obtain upper bounds on $\pi,\sigma,\pi_{2}$ and
$\pi_{u}$ for Table 1. The general framework of flag algebras was developed by
Razborov [58]. Flag algebras has been applied to variety of problems,
including problems on $3$-graphs [2, 1, 58]. By now, it is a standard
application to obtain some upper bounds on $\pi$. For $\sigma,\pi_{2}$ and
$\pi_{u}$ we give a short explanation how upper bounds can be obtained. For
readers familiar with flag algebras, the function counting scaled codegree
squared sum can be expressed using flag algebras as follows:

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}\definecolor[named]{pgffillcolor}{rgb}{1,.75,.75}\pgfsys@color@rgb@fill{1}{.75}{.75}\pgfsys@invoke{
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}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}
{{}{}}{{}{}}{{}} {{{}}{{}}}{{}}{{}{}}{{{}}{{}}}{{}}{}{{}}{}{}{}
{{}{}}{{}{}}{{}} {{{}}{{}}}{{}}{{}{}}{{{}}{{}}}{{}}{}{{}}{}{}{}
{{}{}}{{}{}}{{}}
{{{}}{{}}}{{}}{{}{}}{{{}}{{}}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,.75,.75}\pgfsys@color@rgb@fill{1}{.75}{.75}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@curveto{18.46657pt}{17.29663pt}{15.61469pt}{13.15318pt}{28.45276pt}{28.45276pt}\pgfsys@curveto{15.61469pt}{13.15318pt}{9.98619pt}{11.15613pt}{0.0pt}{28.45276pt}\pgfsys@curveto{14.17924pt}{3.89351pt}{14.27351pt}{24.55925pt}{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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where $(G_{n})_{n\geq 1}$ is a convergent sequence with $G_{n}$ being an
$n$-vertex $3$-graph. For a well-written explanation of the flag algebra
method in the setting of $3$-uniform hypergraphs see [27] and for the
particular application and an explanation of (2) see [3].

For the minimum codegree threshold $\pi_{2}$, we use the formulation from
[26]. Suppose we want to show that $\pi_{2}(H)\leq z$ for some $3$-graph $H$
and $z\in(0,1)$. We do this by arguing that there is no convergent sequence
$(G_{n})_{n\geq 1}$ of $H$-free graphs with minimum codegree $z$. Suppose
there is such a sequence. Then for any flag $F$ with two labeled vertices

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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
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}{}\pgfsys@rect{27.35275pt}{-1.1pt}{2.20001pt}{2.20001pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}-z\right)\times
F\right\rrbracket_{1,2}\geq 0.$ (3)

By combining these inequalities and sum of squares inequalities, one gets a
contradiction with the existence of $(G_{n})_{n\geq 1}$.

It is not obvious to us how to express in the flag algebras language the
conditions for uniform Turán density. Instead of a direct expression, Glebov,
Král’, and Volec [39] formulated consequences of the conditions that can be
expressed using flag algebras. Suppose graphs in a convergent sequence
$(G_{n})_{n\geq 1}$ are $(d,\eta,1)$-dense. The simplest instance is to take
two labeled vertices and look at their co-neighborhood $N$. In $N$, the
relative density of edges must be at least $d$ if $N$ is sufficiently large.
If it is not sufficiently large, then the values in the equation below become
all $0$ anyway. In flag algebras, this can be done as

$\displaystyle\left\llbracket\left((1-d)\vbox{\hbox{ \leavevmode\hbox
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F\right\rrbracket_{1,2}\geq 0,$ (4)

where edges with exactly one labeled vertex are not depicted for the sake of
readability and $F$ is any flag with two labeled vertices. An analogous
equation can be obtained by considering non-co-neighborhood. More generally,
for more labeled vertices, we can decide for each pair if we want to take co-
neighborhood or non-co-neighborhood. The approach by Glebov, Král’, and Volec
[39] continues by getting an exact solution on the threshold that has density
of a particular hypergraph 0. Then to a hypothetical counterexample, one can
apply sparsification to obtain an example on the threshold density, which in
turn shows that the hypothetical counterexample actually has zero edge
density, which is a contradiction. Since the full method from [39] is more
involved and uniform Turán density is not the main focus of this paper, we
opted for a simpler approach. We use (4) and its analogue for non-co-
neighborhood and obtain an upper bound on the global edge density that is
strictly less than $d$, leading to a contradiction. This simplified approach
is not as strong. For $K_{4}^{3-}$ it gives only $0.259$ while a sharp result
0.25 is known [39].

The calculations which lead to the results in Table 1 are computer assisted;
we use CSDP [8] to calculate numerical solutions of semidefinite programs and
then use SageMath [66] for rounding these numerical solutions to exact ones.
The data files and programs to reproduce the calculations we developed are
available at http://lidicky.name/pub/co2b/.
Next, we will present the constructions which give the lower bounds from Table
1.

##  3 Bounds from Table 1

###  3.1 $K_{4}^{3}$

Turán’s tetrahedron problem asks to determine the Turán density of
$K_{4}^{3}$. The best lower bound $\pi(K_{4}^{3})\geq 5/9$ is obtained by
$C_{n}$, see Figure 1, the $3$-graph on $n$ vertices with vertex set
$V(C_{n})=V_{1}\cup V_{2}\cup V_{3}$ where $||V_{i}|-|V_{j}||\leq 1$ for
$i\neq j$ and edge set

$\displaystyle E(C_{n})=\\{abc:a\in V_{1},b\in V_{2},c\in
V_{3}\\}\cup\\{abc:a,b\in V_{1},c\in V_{2}\\}$ $\displaystyle\cup\
\\{abc:a,b\in V_{2},c\in V_{3}\\}\cup\\{abc:a,b\in V_{3},c\in V_{1}\\}.$

Brown [9], Kostochka [49], Fon-der-Flaass [28] and Frohmader [32] constructed
families of $K_{4}^{3}$-free $3$-graphs with the same number of edges. A
series of papers [15, 10, 58] have improved on the upper bound of
$\pi(K_{4}^{3})$ culminating in the current best-known bound by Baber [1]

$\displaystyle\pi(K_{4}^{3})\leq 0.5615.$

We [3] solved Turán’s tetrahedron problem for the codegree squared density by
showing $\sigma(K_{4}^{3})=1/3$, where the lower bound is achieved by $C_{n}$.
For the minimum codegree threshold Czygrinow and Nagle [13] provided a
construction that shows $\pi_{2}(K_{4}^{3})\geq 1/2$. Let $T$ be a uniformly
at random chosen tournament on $n$ vertices. Define a $3$-graph $G_{T}$ on $n$
vertices by setting the triple $ijk$ with $i<j<k$ to be an edge of $G_{T}$ if
the ordered pairs $(i,j)$ and $(i,k)$ receive opposite directions in $T$. This
$3$-graph is $K_{4}^{3}$-free, and has minimum codegree $n/2-o(n)$ with high
probability. Using flag algebras we can prove that $\pi_{2}(K_{4}^{3})\leq
0.529$.

The previous construction was used by Rödl [61] to show that for the uniform
Turán density $\pi_{u}(K_{4}^{3})\geq 1/2$. Erdős [19] suggested that this
might be best possible, also see Reiher [59]. Using flag algebras as described
in Section 2.3, we obtain $\pi_{u}(K_{4}^{3})\leq 0.531$.

###  3.2 $K_{5}^{3}$

For the Turán density of $K_{5}^{3}$ it is known that

$\displaystyle\frac{3}{4}\leq\pi(K_{5}^{3})\leq 0.769533,$

where the upper bound is obtained via flag algebras [68] and the lower bound
is obtained by the balanced complete bipartite 3-graph $B_{n}$ (see Figure 1)
as observed by Turán [67]. The following $n$-vertex $3$-graph $H_{5}$
(presented in [62], see Figure 1) also achieves the lower bound. The vertex
set of $H_{5}$ is divided into $4$ parts $A_{1},A_{2},A_{3},A_{4}$ with
$||A_{j}|-|A_{i}||\leq 1$ for all $1\leq i\leq j\leq 4$ and a triple $e$ is
not an edge of $H_{5}$ iff there is some $j$ ($1\leq j\leq 4$) such that
$|e\cap A_{j}|\geq 2$ and $|e\cap A_{j}|+|e\cap A_{j+1}|=3$, where
$A_{5}=A_{1}$.

We [3] proved that $\sigma(K_{5}^{3})=5/8$, where the lower bound is obtained
by the complete balanced bipartite $3$-graph $B_{n}$. Notice that $H_{5}$ does
not achieve this lower bound.

For the minimum codegree threshold, we have

$\displaystyle\frac{2}{3}\leq\pi_{2}(K_{5}^{3})\leq 0.74,$

where we obtained the upper bound via flag algebras and the lower bound is due
to Falgas-Ravry [24] and Lo and Markström [50] who constructed lower bounds on
the codegree threshold for cliques of arbitrary size $s$. Here we present the
construction due to Falgas-Ravry. Let $c:E(K_{n})\rightarrow[s]$ be a
uniformly at random chosen colouring of the edges of the complete graph on $n$
vertices with $s$ colours. Consider the $3$-graph $G$ based on this colouring
in the following way: A triple $i,j,k$ with $i<j<k$ forms an edge in $G$ if
and only if $c(ij)\neq c(ik)$. This $3$-graph is $K_{s+2}$-free and has
minimum codegree $(1-1/s)n-o(n)$ with high probability. This construction can
be seen as a generalization of the Czygrinow and Nagle construction from the
previous section.

$V_{1}$$V_{2}$$V_{3}$$C_{n}$ $A$$B$$B_{n}$
$A_{1}$$A_{2}$$A_{3}$$A_{4}$$\overline{H_{5}}$

Figure 1: Illustration of $C_{n}$, $B_{n}$ and the complement of $H_{5}$.

For the uniform Turán density, we have

$\displaystyle\frac{2}{3}\leq\pi_{u}(K_{5}^{3})\leq 0.758,$

where the lower bound is obtained from the previously described $3$-graph $G$
and the upper bound from flag algebras.

###  3.3 $K_{6}^{3}$

For the Turán density of $K_{6}^{3}$ we know

$\displaystyle 0.84\leq\pi(K_{6}^{3})\leq 0.8583903,$

where the upper bound was obtained via flag algebras [68] and the lower bound
(see [62]) is obtained by the following $3$-graph $H_{6}$ (see Figure 2).
Divide the vertex set $[n]$ of $H_{6}$ into $5$ parts $A_{1},\ldots,A_{5}$
with $||A_{j}|-|A_{i}||\leq 1$ for all $1\leq i\leq j\leq 5$ and let a triple
$e$ not form an edge of $H_{6}$ iff there is some $j$ ($1\leq j\leq 5$) such
that

$\displaystyle|e\cap A_{j}|\geq 2\quad\quad\text{and}\quad\quad|e\cap
A_{j}|+|e\cap A_{j+1}|=3,$

where $A_{6}=A_{1}$. For the codegree squared density we have

$\displaystyle 0.7348\leq\sigma(K_{6}^{3})\leq 0.7535963,$

where we obtained the upper bound by flag algebras and the lower bound is
achieved by the following construction from [3]. Let $G_{6}$ (see Figure 2) be
a $3$-graph with a vertex partition $A\cup B_{1}\cup B_{2}\cup B_{3}$ such
that $|B_{1}|=|B_{2}|=|B_{3}|=bn$ and $|A|=an$, where $a$ and $b$ are
optimized later. The $3$-graph $G_{6}$ forms a $C_{3bn}$ (see Subsection 3.1)
on $B_{1}\cup B_{2}\cup B_{3}$ and further contains all edges intersecting $A$
with exactly $1$ or $2$ vertices. It has codegree squared sum

$\displaystyle\textup{co}_{2}(G_{6})$
$\displaystyle=\left(\frac{a^{2}}{2}(1-a)^{2}+3ab+3\frac{b^{2}}{2}(a+b)^{2}+3b^{2}(a+2b)^{2}\right)n^{4}(1+o(1))$
$\displaystyle=\left(\frac{11}{18}a^{4}-a^{3}-\frac{2}{3}a^{2}+\frac{8}{9}a+\frac{1}{6}\right)n^{4}(1+o(1)).$

The optimum is one of the roots of $22a^{3}-27a^{2}-12a+8=0$. The root is
approximately $a=0.412498$, which gives
$\textup{co}_{2}(G_{6})=0.3674n^{4}(1+o(1))$. Note that this $3$-graph with
$a=2/5$ is an instance of a family of constructions due to Keevash and Mubayi
[44] that achieves the best-known lower bound for the Turán density of
$K_{6}^{3}$.

For the minimum codegree threshold of $K_{6}^{3}$ we have

$\displaystyle\frac{3}{4}\leq\pi_{2}(K_{6}^{3})\leq 0.838,$

where the upper bound is obtained via flag algebras and the lower bound is due
to Falgas-Ravry [24] and Lo and Markström [50], see the previous section for
the construction by Falgas-Ravry.

For the uniform Turán density, we have

$\displaystyle\frac{3}{4}\leq\pi_{u}(K_{6}^{3})\leq 0.853,$

where the lower bound is obtained by the Falgas-Ravry’s construction stated in
the previous section and the upper bound from flag algebras.

###  3.4 $K_{t}^{k}$

Denote by $K_{t}^{k}$ the complete $k$-graph on $t$ vertices. Large cliques
are studied intensively for all notions of extremality. For an overview of
results in $\ell_{1}$-norm see [63], in $\ell_{2}$-norm see [3], and for the
minimum codegree threshold see [51] and [64].

The best-known bounds for the Turán density are

$\displaystyle 1-\left(\frac{k-1}{t-1}\right)^{k-1}\leq\pi(K_{t}^{k})\leq
1-\binom{t-1}{k-1}^{-1},$

where the lower bound is due to Sidorenko [62] and the upper bound is due to
de Caen [14].

For $k=3$, Keevash and Mubayi [44] constructed a family of $3$-graphs
obtaining this lower bound. Their construction for $K_{t}^{3}$-free graphs
goes as follows. Take a directed graph $G$ on $t-1$ vertices with both in-
degree and out-degree equal to one, i.e. $G$ is a vertex disjoint union of
directed cycles. Note that cycles of length $2$ in $G$ are allowed but loops
are not. Now we construct a $K_{t}^{3}$-free hypergraph $H$ on $(t-1)n$
vertices. Partition $V(H)$ into $A_{1},\ldots,A_{t-1}$ of equal sizes
corresponding to vertices $v_{1},\ldots,v_{t-1}$ of $G$. A triple $xyz$ is not
an edge in $H$ if and only if there exists $A_{i}$ such that $x,y,z\in A_{i}$,
or there exists $v_{i}v_{j}\in E(G)$ with $|\\{x,y,z\\}\cap A_{i}|=2$ and
$|\\{x,y,z\\}\cap A_{j}|=1$. Notice that $H_{5}$ from Figure 1 is a case of
this construction with $G$ being a directed cycle on four vertices with edges
$v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{1}$. Other examples are
$C_{n},B_{n}$ and $H_{6}$.

The best-known extremal constructions in $\ell_{2}$-norm for small $t$ are
possibly unbalanced versions of this construction.

  * •

For $t=4$, $C_{n}$ is the resulting 3-graph when $G$ is a directed triangle.

  * •

For $t=5$, $B_{n}$ is the resulting 3-graph when $G$ is the union of two
cycles of length 2.

  * •

For $t=6$, $G_{6}$ is the resulting 3-graph when $G$ is formed by the union of
a directed triangle with a directed cycle of length 2, up to balancing of the
class sizes.

This suggests [3] that when $G$ is maximizing the number of directed cycles,
the resulting $3$-graph $H$ could be extremal in $\ell_{2}$-norm.

For the minimum codegree threshold of cliques of large size, the best-known
bounds, obtained by Lo and Zhao [51], are

$\displaystyle 1-c_{2}\frac{\log t}{t^{k-1}}\leq\pi_{2}(K_{t}^{k})\leq
1-c_{1}\frac{\log t}{t^{k-1}},$

where $c_{1},c_{2}>0$ are constants depending on $k$.

$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$\overline{H_{6}}$
$B_{2}$$B_{1}$$B_{3}$$A$$\overline{G_{6}}$

Figure 2: Illustration of the complement of $H_{6}$ and $G_{6}$.
12345$F_{3,2}$12345$F_{5}$12345$v_{3}$$v_{2}$$v_{1}$$v_{5}$$v_{4}$$C_{5}^{-}$
Figure 3: Colorings for Theorem 2.2(b).

###  3.5 $F_{3,2}$

Füredi, Pikhurko and Simonovits [35] proved that the Turán density of
$F_{3,2}$ is $4/9$ which is achieved by the $3$-graph with vertex set
$V=V_{1}\cup V_{2}$, where all edges have two vertices in $V_{1}$ and one
vertex in $V_{2}$ and $|V_{1}|=\frac{2}{3}n$.
The $F_{3,2}$-free graph $G$ which we conjecture to maximize the codegree
square sum has similar structure, but different class sizes:
$|V_{1}|=\lceil\sqrt{1/2}n\rceil$. Clearly, changing the class sizes will not
create an $F_{3,2}$, and $\textup{co}_{2}(G)=1/8n^{4}(1+o(1))$. Thus,
$\sigma(F_{3,2})\geq 1/4$. Flag algebras give $\sigma(F_{3,2})\leq
1/4+10^{-9}$. Here we note that flag algebras give a numerical result of $1/4$
but obtaining an exact solution is tricky since it contains $\sqrt{1/2}$ in
addition to rational numbers. Likely, using the stability method one could
prove an exact result, however we have not attempted to do it.
The minimum codegree threshold of $F_{3,2}$ was determined exactly by Falgas-
Ravry, Marchant, Pikhurko and Vaughan [25] who proved that for large enough
$n$

$\displaystyle\textup{ex}_{2}(n,F_{3,2})=\begin{cases}\left\lfloor\frac{n}{3}\right\rfloor-1&\text{
if $n$ is congruent to $1$ modulo $3$,}\\\
\left\lfloor\frac{n}{3}\right\rfloor&\text{ otherwise.}\end{cases}$

The lower bound is obtained by the hypergraph with vertex partition $A_{1}\cup
A_{2}\cup A_{3}$, where $||A_{i}|-|A_{j}||\leq 1$ for $i,j\in[3]$, and edges
$xyz$ for $x,y\in A_{i},z\in A_{i+1}$ for $i=1,2,3$, where $A_{4}:=A_{1}$.

The uniform Turán density of $F_{3,2}$ is $\pi_{u}(F_{3,2})=0$ by Theorem 2.2,
because the edges of the shadow graph can be colored in the following way.
Color the edges $12,34$ red, the edges $13,14,15,35$ blue, and the edges
$23,24,25,45$ green; see Figure 3, and recall that the edges of $F_{3,2}$ are
$123,145,245,345$.

###  3.6 $F_{3,3}$

Keevash and Mubayi [46] and independently Goldwasser and Hansen [40] proved
that the balanced, complete, bipartite $3$-graph is extremal in the
$\ell_{1}$-norm for $n\geq 6$. Thus, $\pi(F_{3,3})=3/4$. In [3] we proved that
it is also extremal in the $\ell_{2}$-norm for $n$ large enough, i.e.
$\sigma(F_{3,3})=5/8$. For the minimum codegree threshold we have
$1/2\leq\pi_{2}(F_{3,3})\leq 0.604$, where the lower bound comes from the
balanced, complete, bipartite $3$-graph $B_{n}$ and the upper bound is
obtained via flag algebras. For the uniform Turán density, we have

$\displaystyle\frac{1}{4}=\pi_{u}(K_{4}^{3-})\leq\pi_{u}(F_{3,3})\leq 0.605,$
(5)

since $K_{4}^{3-}\subset F_{3,3}$. For $\pi_{u}(K_{4}^{3-})$ see Section 3.9.
The upper bound in (5) is obtained by the method of flag algebras.

###  3.7 $F_{5}$

Frankl and Füredi [30] proved that for $n\geq 3000$ the $F_{5}$-free $3$-graph
with the largest number of edges is $S_{n}$, thus, $\pi(F_{5})=2/9$. The
codegree squared density of $F_{5}$ is $\sigma(F_{5})=2/27$ which follows from
Theorem 1.3, see Section 4 for more details.

The minimum codegree threshold of $F_{5}$ is $\pi_{2}(F_{5})=0$. To be more
precise, we have $\textup{ex}_{2}(n,F_{5})\leq 2$, because if there exists an
$F_{5}$-free $3$-graph $G$ with $d(x,y)\geq 3$ for all pairs $xy$, then take
any pair $ab$ and two vertices $c,d\in N(a,b)$. Now there exists $e\neq
a,b,c,d$ with $e\in N(c,d)$. Thus $a,b,c,d,e$ spans an $F_{5}$, a
contradiction.

The uniform Turán density of $F_{5}$ is $\pi_{u}(F_{5})=0$ by Theorem 2.2,
because the edges of the shadow graph can be colored in the following way.
Color the edges $12,34$ red, the edges $13,14,35$ blue, and the edges
$23,24,45$ green; see Figure 3.

###  3.8 Fano plane $\mathbb{F}$

Denote by $\mathbb{F}$ the Fano plane, i.e., the unique $3$-uniform hypergraph
with seven edges on seven vertices in which every pair of vertices is
contained in a unique edge. Sós [65] proposed to study the Turán number for
the Fano plane and conjectured the complete balanced bipartite $3$-graph
$B_{n}$ to be extremal. This problem was solved asymptotically by Caen and
Füredi [16]. Later, Füredi and Simonovits [36] and, independently, Keevash and
Sudakov [47] determined the extremal hypergraph for large $n$. Recently,
Bellmann and Reiher [4] solved the question for all $n$.
We conjecture that the extremal example in the codegree squared sense also is
the complete bipartite graph.

######  Conjecture 3.1.

There exists $n_{0}$, such that for all $n\geq n_{0}$

$\displaystyle\textup{exco}_{2}(n,\mathbb{F})=\textup{co}_{2}(B_{n}).$

Furthermore, $B_{n}$ is the unique $\mathbb{F}$-free $3$-graph $H$ on $n$
vertices satisfying $\textup{co}_{2}(H)=\textup{exco}_{2}(n,\mathbb{F})$.

We have $5/8\leq\sigma(\mathbb{F})\leq 3/4$ as trivial bounds using the
complete bipartite $3$-graph as a lower bound and the fact that
$\sigma(\mathbb{F})\leq\pi(\mathbb{F})\leq 3/4$.

The minimum codegree threshold was asymptotically determined to be
$\pi_{2}(\mathbb{F})=1/2$ by Mubayi [53]. He conjectured that
$\textup{ex}_{2}(n,F_{3,2})=\lfloor n/2\rfloor$ for large enough $n$. This
conjecture was solved by Keevash [43] using an involved quasi-randomness
argument. Later, DeBiasio and Jiang [17] gave a simplified proof.

We did not use flag algebras for $\mathbb{F}$ because our computers cannot
handle the number of $\mathbb{F}$-free $3$-graphs on $7$ vertices.

The uniform Turán density of $\mathbb{F}$ is $\pi_{u}(\mathbb{F})=0$ by
Theorem 2.2, because the edges of the shadow graph can be colored in the
following way. Color the edges $12,34,15,24,14,25,36$ red, the edges
$13,35,16,26,17,27,37$ blue, and the edges $23,45,56,46,47,57,67$ green, where
the edges of the Fano plane are are $123,345,156,246,147,257,367$.

###  3.9 $K_{4}^{3-}$

Denote by $S_{6}$ the $3$-graph on six vertices with edge set

$\displaystyle E(S_{6})=\\{123,234,345,451,512,136,246,356,256,146\\}.$

Let $n=6^{k}$ be a power of $6$ and let $G$ be the iterated blow-up of
$S_{6}$. This $3$-graph $G$ was constructed by Frankl and Füredi [31] to give
a lower bound on the Turán density of $K_{4}^{3-}$. Since $G$ is
$K_{4}^{3-}$-free, $\pi(K_{4}^{3-})\geq 2/7$. The best know upper bound
$\pi(K_{4}^{3-})\leq 0.28689$ is by Vaughan [68] using flag algebras. The
$3$-graph $G$ has codegree squared sum

$\displaystyle\textup{co}_{2}(G)$
$\displaystyle=\frac{1}{2}n^{4}\left(\sum_{i\geq
1}\frac{5}{6^{i}}\left(\frac{2}{6^{i}}\right)^{2}+o(1)\right)=10n^{4}\left(\sum_{i\geq
1}\frac{1}{216^{i}}+o(1)\right)$
$\displaystyle=10n^{4}\left(\frac{1}{1-\frac{1}{216}}-1\right)(1+o(1))=\frac{2}{43}n^{4}(1+o(1)).$

Thus, $\sigma(K_{4}^{3-})\geq 4/43$. We conjecture that $G$ is the extremal
example in $\ell_{2}$-norm.

######  Conjecture 3.2.

$\displaystyle\sigma(K_{4}^{3-})=\frac{4}{43}\approx 0.0930232558.$

Flag algebras only give $\sigma(K_{4}^{3-})\leq 0.09306286$. The minimum
codegree threshold of $K_{4}^{3-}$ was determined to be
$\pi_{2}(K_{4}^{3-})=1/4$ by Falgas-Ravry, Pikhurko, Vaughan and Volec [26]
solving a conjecture by Nagle [56]. The lower bound is obtained by a
construction originally due to Erdős and Hajnal [21]. Given a tournament $T$
on the vertex set $[n]$, define a $3$-graph $C(T)$ on $[n]$ by taking the edge
set to be all triples of vertices inducing a cyclically oriented triangle in
$T$. No tournament on $4$ vertices can contain more than two cyclically
oriented triangles, hence $C(T)$ is $K_{4}^{3-}$-free. If $T$ is chosen
uniformly at random, then the minimum codegree of $C(T)$ is $n/4-o(n)$ with
high probability.

The uniform Turán density of $K_{4}^{3-}$ was determined to be
$\pi_{u}(K_{4}^{3-})=1/4$ by Glebov, Král’ and Volec [39]. The lower bound is
also achieved by $C(T)$.

###  3.10 {$K_{4}^{3-},F_{3,2},C_{5}$}

Denote by $H_{7}$ the $3$-graph on seven vertices with edge set

$\displaystyle
E(H_{7})=\\{124,137,156,235,267,346,457,653,647,621,542,517,431,327\\}.$

Falgas-Ravry and Vaughan [27] proved that
$\sigma(K_{4}^{3-},F_{3,2},C_{5})=12/49$, where the lower bound is achieved by
the blow-up of $H_{7}$ on $n$ vertices. However, the codegree squared density
is achieved by a different $3$-graph, namely the complete balanced $3$-partite
$3$-graph $S_{n}$. The $3$-graph $S_{n}$ is $K_{4}^{3-}$-free, $F_{3,2}$-free
and $C_{5}$-free and it has codegree squared sum
$\textup{co}_{2}(S_{n})=1/27n^{4}$. Thus,
$\sigma(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\geq 2/27$. Flag algebras give
$\sigma(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\leq 2/27$, hence,

$\displaystyle\sigma(\\{K_{4}^{3-},F_{3,2},C_{5}\\})=\frac{2}{27}.$

For the minimum codegree threshold we have

$\displaystyle\frac{1}{12}\leq\pi_{2}(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\leq
0.186,$

where the upper bound is obtained by flag algebras and the lower bound by the
following construction. Consider the intersection of the Erdős-Hajnal [21]
random tournament construction $C(T)$ with the $3$-graph having vertex
partition $A_{1}\cup A_{2}\cup A_{3}$, where $||A_{i}|-|A_{j}||\leq 1$ for
$i,j\in[3]$, and edges $xyz$ for $x,y\in A_{i},z\in A_{i+1}$ for $i=1,2,3$,
where $A_{4}=A_{1}$. This $3$-graph has minimum codegree $n/12-o(n)$ with high
probability, and is $C_{5}$, $F_{3,2}$ and $K_{4}^{3-}$-free.

For the uniform Turán density we have

$\displaystyle\pi_{u}(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\leq\pi_{u}(F_{3,2})=0.$

###  3.11 {$K_{4}^{3-},F_{3,2}$}

Let $G$ be the blow-up of $H_{6}$ on $n$ vertices. This $3$-graph $G$ is
$K_{4}^{3-}$ and $F_{3,2}$-free. Falgas-Ravry and Vaughan [27] proved that
$\pi(K_{4}^{3-},F_{3,2})=5/18$ with the lower bound obtained by $G$. The
codegree squared sum of $G$ is

$\displaystyle\textup{co}_{2}(G)=\binom{6}{2}\left(\frac{n}{6}\right)^{2}\left(\frac{n}{3}\right)^{2}=\frac{5}{108}n^{4}.$

Thus, $\sigma(\\{K_{4}^{3-},F_{3,2}\\})\geq 5/54$. Flag algebras give
$\sigma(\\{K_{4}^{3-},F_{3,2}\\})\leq 5/54$, hence

$\displaystyle\sigma(\\{K_{4}^{3-},F_{3,2}\\})=\frac{5}{54}.$

For the minimum codegree threshold we have

$\displaystyle\frac{1}{12}\leq\pi_{2}(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\leq\pi_{2}(\\{K_{4}^{3-},F_{3,2}\\}\leq
0.202,$

where the upper bound comes from flag algebras. For the uniform Turán density
we have

$\displaystyle\pi_{u}(\\{K_{4}^{3-},F_{3,2}\\})\leq\pi_{u}(F_{3,2})=0.$

###  3.12 {$K_{4}^{3-},C_{5}$}

Denote by $G$ be the iterated blow-up of an edge on $n$ vertices. In other
words, it is obtained from a complete balanced $3$-partite $3$-graph by
inserting a complete balanced $3$-partite $3$-graph in each of the parts
iteratively. The $3$-graph $G$ is $K_{4}^{3-}$ and $C_{5}$-free. This
construction gives the current best lower bound on the Turán density of
$\\{K_{4}^{3-},C_{5}\\}$, see [27]. For the codegree squared density, it gives
$\sigma(\\{K_{4}^{3-},C_{5}\\})\geq{1}/{13}$. We conjecture it to be the
extremal example in $\ell_{2}$-norm.

######  Conjecture 3.3.

$\displaystyle\sigma(\\{K_{4}^{3-},C_{5}\\})=\frac{1}{13}\approx
0.0769230769.$

Flag algebras give $\sigma(\\{K_{4}^{3-},C_{5}\\})\leq 0.07694083$ for the
codegree squared density. For the minimum codegree threshold we have

$\displaystyle\frac{1}{12}\leq\pi_{2}(\\{K_{4}^{3-},F_{3,2},C_{5}\\})\leq\pi_{2}(\\{K_{4}^{3-},C_{5}\\}\leq
0.202,$

where the upper bound comes from flag algebras. For the uniform Turán density,
we have

$\displaystyle\frac{1}{27}\leq\pi_{u}(\\{K_{4}^{3-},C_{5}\\})\leq 0.251,$

where the upper bound is obtained via the method of flag algebras and the
lower bound comes from Theorem 2.2 and Corollary 2.3 which can easily be
modified to also hold for families of $3$-graphs. We check in Section 3.17
that $C_{5}$ does not satisfy property (b) from Theorem 2.2. Since
$\pi_{u}(K_{4}^{3-})=\frac{1}{4}$, it also does not satisfy property (b).

###  3.13 {$F_{3,2},J_{4}$}

Let $G$ be the blow-up of $K_{4}^{3}$ on $n$ vertices. This graph $G$ is
$F_{3,2}$-free and $J_{4}$-free. Falgas-Ravry and Vaughan [27] proved that
$\pi(\\{F_{3,2},J_{4}\\})=3/8$, where the lower bound is achieved by $G$. The
$3$-graph $G$ has codegree squared sum $\textup{co}_{2}(G)=3/32n^{4}(1+o(1))$,
thus, $\sigma(\\{F_{3,2},J_{4}\\})\geq 3/16$. Flag algebras give
$\sigma(\\{F_{3,2},J_{4}\\})\leq 3/16$, hence

$\displaystyle\sigma(F_{3,2},J_{4})=\frac{3}{16}.$

For the minimum codegree threshold we have

$\displaystyle\frac{1}{12}\leq\pi_{2}(\\{K_{4}^{3-},F_{3,2}\\})\leq\pi_{2}(\\{F_{3,2},J_{4}\\}\leq\pi_{2}(F_{3,2})\leq\frac{1}{3},$

because $K_{4}^{3-}$ is a subhypergraph of $J_{4}$. For the uniform Turán
density we have

$\displaystyle\pi_{u}(\\{F_{3,2},J_{4}\\})\leq\pi_{u}(F_{3,2})=0.$

###  3.14 {$F_{3,2},J_{5}$}

Falgas-Ravry and Vaughan [27] proved that $\pi(\\{F_{3,2},J_{5}\\})=3/8$,
where the blow-up of $K_{4}^{3}$ on $n$ vertices achieves the lower bound.
Since $J_{4}\subset J_{5}$, we have

$\displaystyle\frac{3}{16}\leq\sigma(\\{F_{3,2},J_{4}\\})\leq\sigma(\\{F_{3,2},J_{5}\\})\quad\quad\text{and}\quad\quad\frac{1}{12}\leq\pi_{2}(\\{F_{3,2},J_{4}\\})\leq\pi_{2}(\\{F_{3,2},J_{5}\\}).$

By flag algebras we have for the codegree squared density
$\sigma(\\{F_{3,2},J_{5}\\})\leq\frac{3}{16}$ and thus
$\sigma(\\{F_{3,2},J_{5}\\})=3/16$, and for the minimum codegree threshold
$\pi_{2}(\\{F_{3,2},J_{5}\\})\leq 0.28$. For the uniform Turán density we have

$\displaystyle\pi_{u}(\\{F_{3,2},J_{5}\\})\leq\pi_{u}(F_{3,2})=0.$

###  3.15 $J_{4}$

Let $G$ be the iterated blow-up of the complement of the Fano plane on $n$
vertices. This $3$-graph is $J_{4}$-free and conjectured to be an
asymptotically sharp example in $\ell_{1}$-norm, see [6]. It has codegree
squared sum of

$\displaystyle\textup{co}_{2}(G)=\frac{48}{343}n^{4}\sum\left(\frac{1}{7^{4}}\right)^{i}=\frac{7}{50}n^{4}.$

Thus $\sigma(J_{4})\geq 0.28$. We conjecture it to be the extremal example in
$\ell_{2}$-norm.

######  Conjecture 3.4.

$\displaystyle\sigma(J_{4})=0.28.$

Flag algebras only give $\sigma(J_{4})\leq 0.28079696$. For the minimum
codegree threshold and the uniform Turán density of $J_{4}$ we have

$\displaystyle\frac{1}{4}\leq\pi_{2}(K_{4}^{3-})\leq\pi_{2}(J_{4})\leq
0.473\quad\quad\quad\text{and}\quad\quad\quad\frac{1}{4}\leq\pi_{u}(K_{4}^{3-})\leq\pi_{u}(J_{4})\leq
0.49,$

where the upper bounds are obtained using flag algebras and the lower bound
bound holds because $K_{4}^{3-}\subset J_{4}$.

###  3.16 $J_{5}$

Since $J_{4}\subset J_{5}$, we get the following lower bounds:

$\displaystyle\frac{1}{2}\leq\pi(J_{4})\leq\pi(J_{5}),\
0.28\leq\sigma(J_{4})\leq\sigma(J_{5}),\
\frac{1}{4}\leq\pi_{2}(J_{4})\leq\pi_{2}(J_{5})\quad\text{and}\quad\frac{1}{4}\leq\pi_{u}(J_{4})\leq\pi_{u}(J_{5}).$

Flag algebras give us the following upper bounds

$\displaystyle\pi(J_{5})\leq 0.64474184,\quad\quad\sigma(J_{5})\leq
0.4427457,\quad\quad\pi_{2}(J_{5})\leq
0.613\quad\quad\text{and}\quad\quad\pi_{u}(J_{5})\leq 0.631.$

It would be interesting, if one could separate the parameters of $J_{4}$ and
$J_{5}$ from each other.

###  3.17 $C_{5}$

Denote by $G(n)$ the $3$-graph on $n$ vertices where the vertex set is
partitioned into two sets $A$ and $B$ of sizes $bn$ and $(1-b)n$, where
$b\in(0,1)$ and $E(G(n))$ consists of all triples $xyz$, where $x,y\in B$ and
$z\in A$. This $3$-graph is $C_{5}$-free. Let $G_{it}$ be the $3$-graph
constructed from $G(n)$ by iteratively adding $G(bn)$ inside the class $B$.
The $3$-graph $G_{it}$ is also $C_{5}$-free. This construction gives the
current best-known lower bound on the Turán density of $C_{5}$, see [54]. It’s
codegree squared sum can be lower bounded as follows

$\displaystyle\textup{co}_{2}(G_{it})>\left(b^{2}\frac{(1-b)^{2}}{2}+b(1-b)b^{2}\right)\left(\frac{1}{1-(1-b)^{4}}\right)n^{4}>0.12597n^{4}$

for $b$ of size about $0.7$, thus $\sigma(C_{5})\geq 0.25194$. Flag algebras
give $\sigma(C_{5})\leq 0.25310457$. For the minimum codegree threshold of
$C_{5}$ we have

$\displaystyle\frac{1}{3}\leq\pi_{2}(C_{5})\leq 0.3993,$

where the upper bound comes from flag algebras and the lower bound from the
graph with vertex partition $A_{1}\cup A_{2}\cup A_{3}$, where
$||A_{i}|-|A_{j}||\leq 1$ for $i,j\in[3]$, and edges $xyz$ for $x,y\in
A_{i},z\in A_{i+1}$ for $i=1,2,3$, where $A_{4}=A_{1}$. For the uniform Turán
density of $C_{5}$ we have

$\displaystyle\frac{1}{27}\leq\pi_{u}(C_{5})\leq 0.402,$

where the upper bound is obtained via flag algebras and the lower bound by the
following argument. Let $v_{1},v_{2},v_{3},v_{4},v_{5}$ be the vertices of
$C_{5}$, that is
$v_{1}v_{2}v_{3},v_{2}v_{3},v_{4},v_{3}v_{4}v_{5},v_{4}v_{5}v_{1},v_{5}v_{1}v_{2}$
are edges. Using Theorem 2.2, we will show that there exists no vertex
ordering satisfying property (b) from Theorem 2.2 and thus $\pi_{u}(C_{5})\neq
0$. Corollary 2.3 implies $\pi_{u}(C_{5})\geq 1/27$. Without loss of
generality, we can assume that $v_{2}<v_{3}$. We perform a case analysis
depending on the positioning of $v_{4}$.
Case 1: $v_{2}<v_{3}<v_{4}$: Since $v_{2}v_{3}v_{4}$ and $v_{3}v_{4}v_{5}$ are
edges, $v_{5}<v_{3}<v_{4}$. Since $v_{1}v_{2}v_{3}$ and $v_{2}v_{3}v_{4}$ are
edges, $v_{2}<v_{3}<v_{1}$. Because $v_{3}v_{4}v_{5}$ and $v_{4}v_{5}v_{1}$
are edges, we further have $v_{5}<v_{1}<v_{4}$. Since $v_{1}v_{2}v_{5}$ and
$v_{1}v_{2}v_{3}$ are edges, we get $v_{2}<v_{5}<v_{1}$. Thus,
$v_{2}<v_{5}<v_{3}<v_{1}<v_{4}$, contradicting that $v_{5}v_{1}v_{2}$ and
$v_{4}v_{5}v_{1}$ are edges.
Case 2: $v_{4}<v_{2}<v_{3}$: Since $v_{2}v_{3}v_{4}$ and $v_{3}v_{4}v_{5}$ are
edges, $v_{4}<v_{5}<v_{3}$. Since $v_{1}v_{2}v_{3}$ and $v_{2}v_{3}v_{4}$ are
edges, $v_{1}<v_{2}<v_{3}$. Because $v_{3}v_{4}v_{5}$ and $v_{4}v_{5}v_{1}$
are edges, we further have $v_{4}<v_{5}<v_{1}$. Thus we have
$v_{4}<v_{5}<v_{1}<v_{2}<v_{3}$, contradicting that $v_{1}v_{2}v_{5}$ and
$v_{1}v_{2}v_{3}$ are edges.
Case 3: $v_{2}<v_{4}<v_{3}$: Since $v_{2}v_{3}v_{4}$ and $v_{3}v_{4}v_{5}$ are
edges, $v_{5}<v_{4}<v_{3}$. Since $v_{1}v_{2}v_{3}$ and $v_{2}v_{3}v_{4}$ are
edges, $v_{2}<v_{1}<v_{3}$. Further, since $v_{1}v_{2}v_{5}$ and
$v_{1}v_{2}v_{3}$ are edges $v_{2}<v_{1}<v_{5}$. Thus, we have
$v_{2}<v_{1}<v_{5}<v_{4}<v_{3}$, contradicting that $v_{4}v_{5}v_{1}$ and
$v_{5}v_{1}v_{2}$ are edges.

We conclude that there exists no vertex ordering satisfying property (b) from
Theorem 2.2.

###  3.18 $C_{5}^{-}$

Denote by $G$ be the iterated blow-up of an edge on $n$ vertices. The
$3$-graph $G$ is $C_{5}^{-}$-free. This construction gives the current best-
known lower bound on the Turán density of $C_{5}^{-}$, see [54]. For the
codegree squared density, it gives $\sigma(C_{5}^{-})\geq{1}/{13}$. We
conjecture it to be the extremal example in $\ell_{2}$-norm.

######  Conjecture 3.5.

$\displaystyle\sigma(C_{5}^{-})=\frac{1}{13}\approx 0.0769230769.$

Flag algebras give $\sigma(C_{5}^{-})\leq 0.07725405$ for the codegree squared
density and $\pi_{2}(C_{5}^{-})\leq 0.136$ for the minimum codegree threshold.
The uniform Turán density of $C_{5}^{-}$ is $\pi_{u}(C_{5}^{-})=0$. This was
observed by Reiher, Rödl and Schacht [60] using Theorem 2.2. For the sake of
completeness, we repeat the proof here. Let $v_{1},v_{2},v_{3},v_{4},v_{5}$
form a $C_{5}^{-}$, that is
$v_{1}v_{2}v_{3},v_{2}v_{3}v_{4},v_{3}v_{4}v_{5},v_{4}v_{5}v_{1}$ are edges.
Consider the ordering $v_{3}<v_{2}<v_{1}<v_{5}<v_{4}$ of the vertices. The
edges of the shadow graph can be colored in the following way. Color the edges
$v_{3}v_{2},v_{3}v_{5},v_{1}v_{5}$ red, the edges
$v_{1}v_{3},v_{3}v_{4},v_{1}v_{4}$ blue, and the edges
$v_{1}v_{2},v_{2}v_{4},v_{4}v_{5}$ green; see Figure 3. Thus, property (b) of
Theorem 2.2 holds.

###  3.19 $K_{5}^{=}$

Let $K_{5}^{=}$ be the $3$-graph on $5$ vertices with $8$ edges, where the two
missing edges intersect in exactly one vertex. We have

$\displaystyle 0.58656\leq\frac{227}{387}\leq\pi(K_{5}^{=})\leq
0.60961944\quad\text{and}\quad
0.357942\leq\frac{2909}{8127}\leq\sigma(K_{5}^{=})\leq 0.388725.$

The hypergraph providing both lower bounds is the following. Take the
$3$-uniform hypergraph on $n$ vertices with vertex set $V_{1}\cup V_{2}\cup
V_{3}$ such that $||V_{i}|-|V_{j}||\leq 1$ for $i\neq j$ and edge set

$\displaystyle\\{abc:a\in V_{1},b\in V_{2},c\in V_{3}\\}\cup\\{abc:a,b\in
V_{1},c\in V_{2}\\}$ $\displaystyle\cup\ \\{abc:a,b\in V_{2},c\in
V_{3}\\}\cup\\{abc:a,b\in V_{3},c\in V_{1}\\}.$

Now, place in each of the three classes an iterated blowup of $H_{6}$, recall
$H_{6}$ was defined in Section 3.9. This $3$-graph has

$\displaystyle\frac{5}{54}n^{3}+30\sum\left(\frac{n}{3}\cdot\frac{1}{6^{i}}\right)^{3}+o(n^{3})=\frac{227}{2322}n^{3}(1+o(1))\geq
0.58656\binom{n}{3}(1+o(1))$

edges, codegree squared sum of

$\displaystyle\frac{4}{27}n^{4}+3\frac{1}{2}\left(\frac{n}{3}\right)^{4}\sum_{k\geq
1}\frac{5}{6^{k}}\left(\frac{2}{6^{k}}+1\right)^{2}=\frac{2909}{8127}\frac{n^{4}}{2},$

and is $K_{5}^{=}$-free. We conjecture the lower bounds to be asymptotically
sharp.

######  Conjecture 3.6.

$\displaystyle\pi(K_{5}^{=})=\frac{227}{387}\quad\quad\text{and}\quad\quad\sigma(K_{5}^{=})=\frac{2909}{8127}.$

For the minimum codegree threshold and uniform Turán density of $K_{5}^{=}$,
we have

$\displaystyle\frac{1}{2}\leq\pi(K_{4}^{3})\leq\pi_{2}(K_{5}^{=})\leq
0.569\quad\quad\text{and}\quad\quad\frac{1}{2}\leq\pi_{u}(K_{4}^{3})\leq\pi_{u}(K_{5}^{=})\leq
0.567,$

where the upper bound is obtained via flag algebras. It would be interesting
to decide if those parameters are equal to each other.

###  3.20 $K_{5}^{<}$

Let $K_{5}^{<}$ be the $3$-graph on $5$ vertices with $8$ edges, where the two
missing edges intersect in exactly two vertices. Since $K_{4}^{3}$ is a
subgraph of $K_{5}^{<}$, we have

$\displaystyle\frac{5}{9}\leq\pi(K_{4}^{3})\leq\pi(K_{5}^{<}),\quad\frac{1}{3}\leq\sigma(K_{4}^{3})\leq\sigma(K_{5}^{<}),\quad\frac{1}{2}\leq\pi_{2}(K_{4}^{3})\leq\pi_{2}(K_{5}^{<})$
$\displaystyle\text{and}\quad\quad\frac{1}{2}\leq\pi_{u}(K_{4}^{3})\leq\pi_{u}(K_{5}^{<}).$

Using flag algebras we obtain

$\displaystyle\pi(K_{5}^{<})\leq 0.57704775,\quad\sigma(K_{5}^{<})\leq
0.3402107,\quad\pi_{2}(K_{5}^{<})\leq
0.560\quad\text{and}\quad\pi_{u}(K_{5}^{<})\leq 0.568.$

We conjecture the lower bounds to be sharp.

######  Conjecture 3.7.

$\displaystyle\pi(K_{5}^{<})=\frac{5}{9},\quad\quad\sigma(K_{5}^{<})=\frac{1}{3},\quad\quad\pi_{2}(K_{5}^{<})=\frac{1}{2}\quad\quad\text{and}\quad\quad\pi_{u}(K_{5}^{<})=\frac{1}{2}.$

###  3.21 $K_{5}^{-}$

Let $K_{5}^{-}$ be the $3$-graph on $5$ vertices with $9$ edges. We have

$\displaystyle\frac{227}{387}\leq\pi(K_{5}^{=})\leq\pi(K_{5}^{-}),\quad\frac{2909}{8127}\leq\sigma(K_{5}^{=})\leq\sigma(K_{5}^{-}),\quad\frac{1}{2}\leq\pi_{2}(K_{4}^{3})\leq\pi_{2}(K_{5}^{-})$
$\displaystyle\text{and}\quad\quad\frac{1}{2}\leq\pi_{u}(K_{4}^{3})\leq\pi_{u}(K_{5}^{-}).$

Using flag algebras we obtain

$\displaystyle\pi(K_{5}^{-})\leq 0.64208589,\quad\sigma(K_{5}^{-})\leq
0.41961743,\quad\pi_{2}(K_{5}^{-})\leq
0.621\quad\text{and}\quad\pi_{u}(K_{5}^{-})\leq 0.626.$

It would be interesting to decide if those parameters are equal to each other.

##  4 Proof of Theorem 1.3

Bollobás [5] proved that the $\\{F_{4},F_{5}\\}$-free $3$-graph with the
largest number of edges is $S_{n}$, the complete balanced $3$-partite
$3$-graph. His idea was to construct an almost partition of three pairwise
disjoint sets with the property that every edge has at most one point from
each and then use a counting argument to show that the number of edges is
maximized when the classes form a partition and have equal size. This result
of Bollobás [5] was extended by Frankl and Füredi [30] who proved that for
$n\geq 3000$ the $F_{5}$-free $3$-graph with the largest number of edges is
$S_{n}$. Keevash and Mubayi [45] proved a stability result for $F_{5}$-free
graphs in the $\ell_{1}$-norm using a different method from the one in [5].
There is a straightforward modification of Bollobás’ result which we will use
to get a stability result for $F_{5}$-free graphs in the $\ell_{2}$-norm.

######  Theorem 4.1.

For every $\varepsilon>0$ there exists $\delta>0$ and $n_{0}\in\mathbb{N}$
such that if $G$ is an $F_{5}$-free $3$-uniform hypergraph on $n\geq n_{0}$
vertices with $\textup{co}_{2}(G)\geq(1-\delta)\frac{1}{27}n^{4}$, then we can
partition $V(H)=A\cup B\cup C$ such that

$\displaystyle e(A)+e(B)+e(C)+e(A,B)+e(B,C)+e(A,C)\leq\varepsilon n^{3}$ (6)

and

$\displaystyle e(A,B,C)\geq\frac{1}{27}n^{3}-\varepsilon n^{3}.$ (7)

Note that, unlike the $\ell_{1}$-norm case, here it is not instantly obvious
why (6) and (7) are the same conditions, as having high codegree square sum
does not imply the existence of many edges.

###### Proof.

Let $\varepsilon>0$ and choose a constant $\delta=\delta(\varepsilon)$ such
that the following argument holds. Let $G$ be an $n$-vertex $F_{5}$-free
$3$-graph satisfying $\textup{co}_{2}(G)\geq(1-\delta)1/27n^{4}$. Choose
$v_{1},v_{2}\in V(G)$ such that $d(v_{1},v_{2})$ is maximum among all pairs of
vertices. Denote $A^{\prime}=N(v_{1},v_{2})$ and note that there is no
hyperedge with two vertices from $A^{\prime}$ and one vertex from
$V(G)\setminus\\{v_{1},v_{2}\\}$ because $G$ is $F_{5}$-free, see Figure 4 for
an illustration. Choose $v_{3}\in A^{\prime},v_{4}\in
V(G)\setminus(A^{\prime}\cup\\{v_{1},v_{2}\\})$ such that $d(v_{3},v_{4})$ is
maximum and denote $B^{\prime}=N(v_{3},v_{4})\setminus\\{v_{1},v_{2}\\}$. We
have $A^{\prime}\cap B^{\prime}=\emptyset$, because for $x\in A^{\prime}\cap
B^{\prime}$, $v_{3}v_{4}x$ is an edge with two vertices in $A^{\prime}$ and
one in $V(G)\setminus\\{v_{1},v_{2}\\}$. Again, there are no edges with two
vertices from $B^{\prime}$ and one vertex from
$V(G)\setminus\\{v_{1},v_{2},v_{3},v_{4}\\}$ because $G$ is $F_{5}$-free.
Next, choose $v_{5}\in A^{\prime}\setminus\\{v_{3}\\},\ v_{6}\in
B^{\prime}\setminus\\{v_{1},v_{2},v_{3},v_{4}\\}$ such that $d(v_{5},v_{6})$
is maximum. Set
$C^{\prime}=N(v_{5},v_{6})\setminus\\{v_{1},v_{2},v_{3},v_{4}\\}$. We have
$C^{\prime}\cap A^{\prime}=\emptyset$ and $C^{\prime}\cap
B^{\prime}=\emptyset$ as otherwise there is an edge with two vertices in
either $A^{\prime}$ or $B^{\prime}$ and one in
$V(G)\setminus\\{v_{1},v_{2},v_{3},v_{4}\\}$, which would give a copy of
$F_{5}$. Set $E:=\\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\\}$. There is no edge
with two vertices in $C^{\prime}$ and one in $V(G)\setminus E$, because if
$c_{1}c_{2}x$ is such an edge, then $\\{v_{5},v_{6},c_{1},c_{2},x\\}$ induces
an $F_{5}$. Set

$\displaystyle A:=A^{\prime}\setminus E,\quad B:=B^{\prime}\setminus E,\quad
D=V(G)\setminus(A\cup B\cup C^{\prime}),\quad C=C^{\prime}\cup
D=V(G)\setminus(A\cup B)$

and note that $V(G)=A\cup B\cup C^{\prime}\cup D=A\cup B\cup C$. Denote
$a=|A|/n$, $b=|B|/n$, $c^{\prime}=|C^{\prime}|/n$ and $d=|D|/n$. Thus,
$a+b+c^{\prime}+d=1$.

$v_{1}$$v_{2}$$A^{\prime}$$B^{\prime}$$C^{\prime}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$
Figure 4: $A^{\prime},B^{\prime},C^{\prime}$ and
$v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}$ from the proof of Theorem 4.1.

######  Claim 4.2.

We have

$\displaystyle\left|a-\frac{1}{3}\right|\leq\frac{\varepsilon}{10},\quad\quad\left|b-\frac{1}{3}\right|\leq\frac{\varepsilon}{10},\quad\quad\left|c^{\prime}-\frac{1}{3}\right|\leq\frac{\varepsilon}{10},\quad\quad
d\leq\frac{\varepsilon}{10}.$

###### Proof.

Assume, toward a contradiction, that the claim is false. The only triples with
two vertices from either $A$, $B$ or $C^{\prime}$ which can be edges are those
with one vertex in $E$. Therefore, we get