#  Rainbow and Gallai–Rado numbers involving binary function
equations111Supported by NSFC No.12131013 and 12161141006.

Xueliang Li, Yuan Si
Center for Combinatorics, Nankai University, Tianjin 300071, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Let $\mathcal{E}$, $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be equations, $n$
and $k$ be positive integers. The rainbow number
$\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum number of
colors such that for every exact
$(\operatorname{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a
rainbow solution of $\mathcal{E}$. The Gallai–Rado number
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})$ is defined as the
minimum positive integer $N$, if it exists, such that for all $n\geq N$, every
$k$-colored $[n]$ contains either a rainbow solution of $\mathcal{E}_{1}$ or a
monochromatic solution of $\mathcal{E}_{2}$. In this paper, we get some exact
values of rainbow and Gallai–Rado numbers involving binary function equations.
We also provide an algorithm to calculate the rainbow numbers of nonlinear
binary function equations.
Keywords: Coloring; Rainbow number; Gallai–Rado number; Binary function
equation
AMS subject classification 2020: 05D10, 11B25, 11B75.

##  1 Introduction

Let $\mathbb{N}_{+}=\\{1,2,\ldots\\}$ be the set of positive natural numbers
and $n$ be a positive integer. For convenience, we denote the set
$\\{1,2,\ldots,n\\}$ as $[n]$. A _$k$ -coloring_ of a set $[n]$ is a function
$\chi:[n]\rightarrow[k]$, where $[k]$ is the set of colors. We usually use
different numbers to represent different colors, and when the number of colors
is relatively small, we can use specific color names, such as red, blue,
green, and so on. If $\chi$ is a surjective, then we call the coloring $\chi$
_exact_. In other words, an exact coloring requires that each color be used at
least once. Obviously, if $\chi$ is exact, then the number of colors $k$ does
not exceed the number of elements in the colored set $[n]$, that is, $k\leq
n$. All the colorings we consider in this paper are exact. In this paper, we
use $y=f(x)$ to represent a binary function equation. If $x_{0}$, $y_{0}$ is a
solution of $y=f(x)$, then we usually write the solution as $(x_{0},y_{0})$ or
$(x_{0},f(x_{0}))$.

###  1.1 Schur and Rado numbers

Ramsey theory is an important theory developed in the 20th century. In
particular, Ramsey theory is widely used in graph theory and number theory,
and is currently one of the hot and difficult research areas. In brief, Ramsey
theory states that under sufficiently large structures, there must exist a
substructure with specific properties. The main contribution is attributed to
Ramsey, who published a pioneering paper [18] in 1930. In fact, the study of
Ramsey theory predates 1930. In 1916, Schur gave a theorem, later known as the
Schur’s theorem, which is one of the most important theorems in the early era
of Ramsey theory. For a historical introduction to Ramsey theory, we refer to
the first chapter of the monograph [29].

######  Theorem 1.1.

[28] For each integer $k\in\mathbb{N}_{+}$, there is a positive integer $S(k)$
such that every $k$-colored $[S(k)]$ contains a monochromatic solution of the
equation $x+y=z$.

Generally speaking, if the colors of all numbers in a solution of an equation
are the same, then we call the solution a _monochromatic solution_ ; if the
colors of all numbers in a solution of an equation are different, then we call
the solution a _rainbow solution_.

The $S(k)$ in Theorem 1.1 is called the _Schur number_. To date, the exact
values of the Schur numbers only for $1\leq k\leq 5$ have been determined.
That is, $S(1)=1$, $S(2)=5$, $S(3)=14$, $S(4)=45$ and $S(5)=161$. For details,
we refer to [1, 10].

Rado was one of the best Ph.D students of Schur. In 1933, Rado generalized
Schur’s theorem to general linear equations in his Ph.D thesis [17]. Rado
called a linear equation $\sum_{i=1}^{n}a_{i}x_{i}=b$ _$k$ -regular_ if there
exists a monochromatic solution of the equation in any $k$-colored
$\mathbb{N}_{+}$, where $k$ is a positive integer. If a linear equation
$\sum_{i=1}^{n}a_{i}x_{i}=b$ is $k$-regular for all $k\in\mathbb{N}_{+}$, then
the equation is _regular_. According to the definition of regular equation and
Theorem 1.1, the equation $x+y=z$ is regular. Rado gave a necessary and
sufficient condition for judging whether a homogeneous linear equation is
regular.

######  Theorem 1.2.

[17] A homogeneous linear equation $\sum_{i=1}^{n}a_{i}x_{i}=0$ is regular if
and only if there exists $I\subseteq[n]$ such that $\sum_{i\in I}a_{i}=0$.

Inspired by Rado, the definition of regularity of equations can be extended
beyond linear equations, as shown in the following definition.

######  Definition 1.3.

Let $k$ be a positive integer. An equation $\mathcal{E}$ is $k$-regular if
there exists a monochromatic solution of the equation in any $k$-colored
$\mathbb{N}_{+}$. If an equation $\mathcal{E}$ is $k$-regular for all
$k\in\mathbb{N}_{+}$, then the equation is regular.

As a generalization of Schur number, the following is the definition of Rado
number. Noticing that, unlike Schur number, given a positive integer $k$ and
an equation $\mathcal{E}$, the Rado number of $\mathcal{E}$ may not always
exist.

######  Definition 1.4.

Let $\mathcal{E}$ be an equation and $k\geq 2$ be an integer. The Rado number
$\operatorname{R}_{k}(\mathcal{E})$ is defined as the minimum positive
integer, if it exists, such that every $k$-colored
$[\operatorname{R}_{k}(\mathcal{E})]$ contains a monochromatic solution of
$\mathcal{E}$.

The following observation gives a basic method for proving the upper and lower
bounds of the Rado number.

######  Observation 1.

Let $\mathcal{E}$ be an equation and $k\geq 2$ be an integer. If there exists
an exact $k$-coloring of $[n-1]$ such that there is no monochromatic solution
of $\mathcal{E}$, then $\operatorname{R}_{k}(\mathcal{E})\geq n$. If every
exact $k$-coloring of $[n]$ ensures that there is a monochromatic solution of
$\mathcal{E}$, then $\operatorname{R}_{k}(\mathcal{E})\leq n$. Moreover, if
there exists an exact $k$-coloring of $\mathbb{N}_{+}$ such that there is no
monochromatic solution of $\mathcal{E}$, then
$\operatorname{R}_{k}(\mathcal{E})$ does not exist.

Solving the exact value or upper and lower bounds of the Rado number of an
equation has always been a hot topic. Some results on homogeneous linear
equations were studied in references [9, 11, 13, 19, 21, 22, 23, 24, 26, 30],
some results on non-homogeneous linear equations were studied in references
[2, 4, 14, 16, 20, 25, 27], and some results on nonlinear equations were
studied in references [8, 16]. In fact, there are many other relevant
references, which are not listed here.

###  1.2 Rainbow and Gallai–Rado numbers

In graph theory, a _monochromatic subgraph_ refers to a subgraph where the
colors of all edges are the same, while a _rainbow subgraph_ refers to a
subgraph where the colors of all edges are different. The definition of
rainbow number in graph theory is relatively early, and we refer to the
definition given by Schiermeyer [12] in 2004. However, the study of rainbow
numbers in equations is later than that in graph theory, and we refer to the
definition given by Fallon, Giles, Rehm, Wagner and Warnberg [5] in 2020.

######  Definition 1.5.

[12] Let $G$ be a graph and $n$ be a positive integer. The rainbow number
$\operatorname{rb}(n,G)$ is difined as the minimum number of colors such that
for every exact $(\operatorname{rb}(n,G))$-coloring of complete graph $K_{n}$,
there exists a rainbow subgraph $G$.

######  Definition 1.6.

[5] Let $\mathcal{E}$ be an equation and $n$ be a positive integer. The
rainbow number $\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum
number of colors such that for every exact
$(\operatorname{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a
rainbow solution of $\mathcal{E}$.

Fallon, Giles, Rehm, Wagner and Warnberg also gave the following observation,
which is the basic method for proving the upper and lower bounds of the
rainbow number.

######  Observation 2.

[5] If there exists an exact $(k-1)$-coloring of $[n]$ such that the equation
$\mathcal{E}$ has no rainbow solution in $[n]$, then
$\operatorname{rb}([n],\mathcal{E})\geq k$. If every exact $k$-coloring of
$[n]$ ensures that the equation $\mathcal{E}$ has a rainbow solution in $[n]$,
then $\operatorname{rb}([n],\mathcal{E})\leq k$.

In 1967, Gallai’s paper [7] for the first time revealed the structure of
colored complete graphs without rainbow triangles, and thus gave rise to a new
research direction in graph theory known as the _Gallai–Ramsey number_. In
2010, Faudree, Gould, Jacobson and Magnant in [6] defined Gallai–Ramsey number
$\operatorname{gr}_{k}(G:H)$. For more information about Gallai–Ramsey number,
we refer to the monograph [15].

######  Definition 1.7.

[6] Given two non-empty graphs $G,H$ and a positive integer $k$, define the
Gallai–Ramsey number $\operatorname{gr}_{k}(G:H)$ to be the minimum integer
$N$ such that for all $n\geq N$, every $k$-edge-colored complete graph $K_{n}$
contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$.

Due to the rapid development of the Gallai–Ramsey number in the past decade,
it has become a hot research area in graph theory. Inspired by this problem,
the definition of the _Gallai–Schur number_ was proposed in Budden’s paper [3]
in 2020, introducing this type of problem from graph theory to number theory.
Since the Gallai–Schur number only studies the equation $x+y=z$, it can be
generalized to other equations, and similarly defined as the _Gallai–Rado
number_. It can be said that the Gallai–Rado number is a generalized
definition of the Gallai–Schur number.

######  Definition 1.8.

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be two equations and $k$ be a
positive integer. The Gallai–Rado number
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})$ is defined as the
minimum positive integer $N$, if it exists, such that for all $n\geq N$, every
$k$-colored $[n]$ contains either a rainbow solution of $\mathcal{E}_{1}$ or a
monochromatic solution of $\mathcal{E}_{2}$.

For simplicity, without causing confusion, we sometimes say that
$\mathcal{E}_{1}$ in $\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})$
is the _rainbow equation_ and $\mathcal{E}_{2}$ is the _monochromatic
equation_. Noticing that one of the biggest differences between the
Gallai–Rado number and the Gallai–Ramsey number is that the Gallai–Rado number
does not always exist. Next, we present an observation on the basic method for
proving the upper and lower bounds of the Gallai–Rado number.

######  Observation 3.

If there exists an exact $k$-coloring of $[N-1]$ such that there is neither a
rainbow solution of $\mathcal{E}_{1}$ nor a monochromatic solution of
$\mathcal{E}_{2}$, then
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})\geq N$. If every exact
$k$-coloring of $[n]\,(n\geq N)$ ensures that there is either a rainbow
solution of $\mathcal{E}_{1}$ or a monochromatic solution of
$\mathcal{E}_{2}$, then
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})\leq N$. Moreover, if
there exists an exact $k$-coloring of $\mathbb{N}_{+}$ such that there is
neither a rainbow solution of $\mathcal{E}_{1}$ nor a monochromatic solution
of $\mathcal{E}_{2}$, then
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})$ does not exist.

###  1.3 Main results

In Section $2$, we introduce some new definitions that are important in
Sections $3$ and $4$. For example, the $\lambda$-class defined in Section $2$
is used to describe the colored structure of $[n]$ without rainbow solution of
the equation $y=ax+b$.

In Section $3$, we first give the exact value of the rainbow number of the
binary linear equation $y=ax+b$. Then, for nonlinear binary function
equations, the formulas for their rainbow numbers are not easily given
directly. Therefore, we define a parameter called monochromatic parameter. Due
to the close relationship between the monochromatic parameter we defined and
the rainbow number, we also give an algorithm to solve the monochromatic
parameters of nonlinear binary function equations.

In Section $4$, we first give the general properties and related connections
of the Rado numbers and Gallai–Rado numbers of the binary linear equation
$y=ax+b$, and then we present one of the main results, which gives the exact
values of the Gallai–Rado numbers of the rainbow equation $y=x+b$ versus
monochromatic general multivariate linear equations with the fixed number of
colors $b$. Next, we keep the rainbow equation as $y=x+b$, consider the
monochromatic nonlinear binary function equations, and give the exact values
of the Gallai–Rado numbers.

In Section $5$, as a conclusion, we give the relationship between the rainbow
numbers and the Gallai–Rado numbers of equations, and explain why our study of
the rainbow equation for the Gallai–Rado numbers in this paper is only
$y=x+b$. Finally, we raise some questions and give some prospects.

##  2 Preliminaries

In the following definition, the $\lambda$-class
$\mathcal{C}_{(y=f(x),\lambda)}$ we provide is crucial for some of our
results.

######  Definition 2.1.

Let $y=f(x)$ be a strictly monotonically increasing binary function equation
such that $f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$, and let
integer $n\geq f(1)$. For each $\lambda\in[n]$, we define the $\lambda$-class
of $y=f(x)$ as
$\mathcal{C}_{(y=f(x),\lambda)}=\left\\{\lambda,f(\lambda),f(f(\lambda)),\ldots\right\\}\subseteq[n]$.
If all the numbers in $\mathcal{C}_{(y=f(x),\lambda)}$ are of the same color,
then we call the set $\mathcal{C}_{(y=f(x),\lambda)}$ monochromatic, and the
color of the set $\mathcal{C}_{(y=f(x),\lambda)}$ is the same as the color of
the numbers in it. Moreover, the $\lambda$-class can also be defined as a
subset of $\mathbb{N}_{+}$, that is,
$\mathcal{C}_{(y=f(x),\lambda)}=\left\\{\lambda,f(\lambda),f(f(\lambda)),\ldots\right\\}\subseteq\mathbb{N}_{+}$
for each $\lambda\in\mathbb{N}_{+}$.

######  Lemma 2.2.

Let ingeters $a\geq 1$ and $b\geq 0$. If the recurrence of the sequence
$\\{x_{i}:i\geq 1\\}$ is $x_{i+1}=ax_{i}+b$, then the general term is
$x_{i+1}=a^{i+1}\left(\sum_{j=1}^{i}\frac{b}{a^{j+1}}+\frac{x_{1}}{a}\right)$.

###### Proof.

The recurrence of $\\{x_{i}:i\geq 1\\}$ can be rewritten as

$\begin{split}\frac{x_{i+1}}{a^{i+1}}&=\frac{b}{a^{i+1}}+\frac{x_{i}}{a^{i}}\\\
&=\frac{b}{a^{i+1}}+\frac{b}{a^{i}}+\frac{x_{i-1}}{a^{i-1}}\\\
&\qquad\qquad\vdots\\\
&=\sum_{j=1}^{i}\frac{b}{a^{j+1}}+\frac{x_{1}}{a}.\end{split}$

The result thus follows. ∎

Based on Lemma 2.2 and Definition 2.1, we give an example of the
$\lambda$-class of $y=ax+b$.

######  Example 1.

For integers $a\geq 1$, $b\geq 0$ and $n\geq a+b$, and for each
$\lambda\in[n]$, we have

$\mathcal{C}_{(y=ax+b,\lambda)}=\left\\{a^{i}\left(\sum_{j=1}^{i-1}\frac{b}{a^{j+1}}+\frac{\lambda}{a}\right):i\in\mathbb{N}_{+}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
\sum_{j=1}^{0}\frac{b}{a^{j+1}}=0\right\\}\subseteq[n].$

Next, we present the colored structure theorem for $[n]$ without rainbow
solution of $y=f(x)$.

######  Theorem 2.3.

Let $y=f(x)$ be a strictly monotonically increasing binary function equation
such that $f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$, and let
integer $n\geq f(1)$. A colored set $[n]$ contains no rainbow solution of
$y=f(x)$ if and only if for each $\lambda\in[n]$, the $\lambda$-class
$\mathcal{C}_{(y=f(x),\lambda)}\subseteq[n]$ is monochromatic.

###### Proof.

Firstly, we assume that for each $\lambda\in[n]$, the $\lambda$-class
$\mathcal{C}_{(y=f(x),\lambda)}\subseteq[n]$ is monochromatic. Let
$(x_{0},y_{0})$ be an arbitrary solution of the binary function equation
$y=f(x)$ in $[n]$. Since
$x_{0},y_{0}\in\mathcal{C}_{(y=f(x),x_{0})}=\\{x_{0},f(x_{0}),f(f(x_{0})),\ldots\\}\subseteq[n]$,
it follows that $[n]$ contains no rainbow solution of $y=f(x)$.

Next, we assume that the colored set $[n]$ contains no rainbow solution of
$y=f(x)$. Thus, for each $\lambda\in[n]$, $\lambda$ and $f(\lambda)$ are of
the same color. Similarly, $f(\lambda)$ and $f(f(\lambda))$ must also be of
the same color. According to the recursion, we get all the numbers in
$\left\\{\lambda,f(\lambda),f(f(\lambda)),\ldots\right\\}\subseteq[n]$ are of
the same color, which implies that the $\lambda$-class
$\mathcal{C}_{(y=f(x),\lambda)}\subseteq[n]$ is monochromatic. The result thus
follows. ∎

In order to maximize the number of colors used without the rainbow solution of
$y=f(x)$ in the colored set $[n]$, based on Theorem 2.3, we construct a
coloring of $[n]$ as follows.

######  Definition 2.4.

Let $\chi$ be a coloring of $[n]$ as follows: For each $\lambda\in[n]$, the
$\lambda$-class $\mathcal{C}_{(y=f(x),\lambda)}\subseteq[n]$ is monochromatic
and for each pair of different $\lambda_{i}$ and $\lambda_{j}$ satisfies
$\mathcal{C}_{(y=f(x),\lambda_{i})}\cap\mathcal{C}_{(y=f(x),\lambda_{j})}=\emptyset$,
$\mathcal{C}_{(y=f(x),\lambda_{i})}$ and $\mathcal{C}_{(y=f(x),\lambda_{j})}$
are of different colors.

According to Theorem 2.3, the coloring $\chi$ of $[n]$ given in Definition 2.4
is the coloring that maximize number of colors used and ensure that $y=f(x)$
has no rainbow solution.

######  Corollary 2.5.

Let ingeters $b\geq 2$ and $n\geq b+1$. A $b$-colored set $[n]$ contains no
rainbow solution of $y=x+b$ if and only if for each $\lambda\in[b]$, the
$\lambda$-class $\mathcal{C}_{(y=x+b,\lambda)}\subseteq[n]$ is monochromatic.
Furthermore, for different $\lambda_{i}$ and $\lambda_{j}$ in $[b]$,
$\mathcal{C}_{(y=x+b,\lambda_{i})}$ and $\mathcal{C}_{(y=x+b,\lambda_{j})}$
have different colors.

##  3 Results for rainbow numbers

######  Theorem 3.1.

For ingeters $a\geq 1$, $b\geq 0$ and $n\geq a+b$, we have

$\operatorname{rb}([n],y=ax+b)=n-\left\lfloor\frac{n-b}{a}\right\rfloor+1.$

###### Proof.

For the lower bound, we only need to construct a
$\left(n-\left\lfloor\frac{n-b}{a}\right\rfloor\right)$-coloring of $[n]$, so
that the equation $y=ax+b$ has no rainbow solution in $[n]$. Actually, the
coloring $\chi$ of $[n]$ we constructed is follows from Definition 2.4. The
specific steps for constructing coloring are given in Algorithm 1.

Input: Ingeters $a\geq 1$, $b\geq 0$ and $n\geq a+b$.

Output: A colored set $[n]$ without rainbow solution of $y=ax+b$.

1

2Let $t=1$ and $S=\emptyset$;

3 while  _$t\leq n$_ do

4 if  _$t\notin\mathcal{C}_{(y=ax+b,\lambda)}$ for all $\lambda\in S$_ then

5 assign a new color to $t$;

6 $S=S\cup\\{t\\}$;

7 $t=t+1$;

8

9 else

10 the color assigned to $t$ is the same as the color of
$\mathcal{C}_{(y=ax+b,\lambda_{0})}$, where
$t\in\mathcal{C}_{(y=ax+b,\lambda_{0})}$;

11 $t=t+1$;

12

13 end if

14

15 end while

Algorithm 1 The coloring $\chi$ of $[n]$.

Next, we calculate how many colors are used for coloring $\chi$. According to
recursion, for each $\lambda\in[n]$, each number in
$\mathcal{C}_{(y=ax+b,\lambda)}\setminus\\{\lambda\\}$ can be written as
$ia+b\,(i\in\mathbb{N}_{+})$, so all numbers in the subset
$\\{ia+b:i\in\mathbb{N}_{+}\\}\subseteq[n]$ cannot be assigned new colors.
Therefore, we only need to calculate the number of numbers in $[n]$ that
cannot be assigned new colors. For the convenience of counting, we list the
following table. Numbers with lightgray boxes in the table cannot be assigned
new colors, while other numbers are assigned different colors. It is easy to
see that there are $\left\lfloor\frac{n-b}{a}\right\rfloor$ numbers that
cannot be assigned new colors, that is, the coloring $\chi$ uses
$n-\left\lfloor\frac{n-b}{a}\right\rfloor$ colors.

Count the number of colors used for coloring $\chi$
---
$1$ | … | $b$ | $b+1$ | … | $a+b-1$ | $a+b$
& … | ⋮  | … | …  | $n$ | … | ⋮

For the upper bound, we arbitrarily color $[n]$ with
$n-\left\lfloor\frac{n-b}{a}\right\rfloor+1$ colors. Recall that if there is
no rainbow solution of $y=ax+b$ in $[n]$, then according to Theorem 2.3, each
$\lambda$-class is monochromatic. Noticing that under the coloring $\chi$
constructed above, all the numbers in $\mathcal{C}_{(y=ax+b,\lambda)}$ have
the same color for each $\lambda\in[n]$, and if each pair of different
$\lambda_{i}$ and $\lambda_{j}$ satisfies
$\mathcal{C}_{(y=f(x),\lambda_{i})}\cap\mathcal{C}_{(y=f(x),\lambda_{j})}=\emptyset$,
$\mathcal{C}_{(y=ax+b,\lambda_{i})}$ and $\mathcal{C}_{(y=ax+b,\lambda_{j})}$
are of different colors. We have counted
$n-\left\lfloor\frac{n-b}{a}\right\rfloor$ colors used for the coloring
$\chi$, and if an additional color is added, then according to pigeonhole
principle, it will inevitably lead to the existence of an integer
$\lambda_{0}\in[n]$, resulting in two numbers with different colors in
$\mathcal{C}_{(y=ax+b,\lambda_{0})}$, which implies that there is a rainbow
solution of $y=ax+b$ in $[n]$. The result thus follows. ∎

We consider the rainbow number of binary function equations. Assuming that
strictly monotonically increasing binary function equations satisfy
$f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$. Thus, there is an
inverse function $x=f^{-1}(y)$ for $y=f(x)$, and if $y$ is not a positive
integer, then $f^{-1}(y)$ must also be not a positive integer. If $y=f(x)$ has
no solution in $[n]$, then naturally there is no rainbow solution. Therefore,
it is meaningful to study the rainbow number of $y=f(x)$ only when $y=f(x)$
has at least one solution in $[n]$, that is, $n\geq f(1)$. Now we know that
$y=f(x)$ has a solution in $[n]$, and the natural idea is that if all numbers
in $[n]$ are assigned different colors, then there must be a rainbow solution
of $y=f(x)$, but in this case, the number of colors is the largest. Next, we
try to color some of the numbers in $[n]$ the same color in an attempt to
reach the extreme value. Definition 3.2 is to define this extreme value
characteristic.

######  Definition 3.2.

Let $y=f(x)$ be a strictly monotonically increasing binary function equation
such that $f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$, and integers
$n\geq f(1)$ and $\mu\geq 1$. If there exists a $(n-\mu)$-coloring of $[n]$
such that $y=f(x)$ has no rainbow solution in $[n]$, but for all
$(n-\mu+1)$-coloring of $[n]$, $y=f(x)$ always has a rainbow solution in
$[n]$, then we call $\mu$ the monochromatic parameter with respect to $y=f(x)$
and $n$.

Based on Theorem 3.1 and Definition 3.2, we give an example of the
monochromatic parameter $\mu$ with respect to $y=ax+b$ and $n$.

######  Example 2.

Let integers $a\geq 1$, $b\geq 0$ and $n\geq a+b$. The monochromatic parameter
with respect to $y=ax+b$ and $n$ is

$\mu=\left\lfloor\frac{n-b}{a}\right\rfloor.$

Combining Definition 3.2 and Observation 2, we directly provide the following
observation.

######  Observation 4.

Let $y=f(x)$ be a strictly monotonically increasing binary function equation
such that $f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$, and integer
$n\geq f(1)$. If $\mu$ is the monochromatic parameter with respect to $y=f(x)$
and $n$, then

$\operatorname{rb}([n],y=f(x))=n-\mu+1.$

We can obtain the explicit expression for the monochromatic parameter $\mu$
with respect to $y=ax+b$ and $n$ due to the linear properties of $y=ax+b$.
However, for a nonlinear binary function equation $y=f(x)$ that satisfies
strictly monotonically increasing and $f(x)\in\mathbb{N}_{+}$ for all
$x\in\mathbb{N}_{+}$, it is not easy to directly obtain the explicit
expression of its monochromatic parameter $\mu$. The Algorithm 2 we provide
next can be applied to find the monochromatic parameters of the general
nonlinear binary function equations.

Input: A strictly monotonically increasing binary function equation $y=f(x)$
such that $f(1)\geq 2$ and $f(x)\in\mathbb{N}_{+}$ for all
$x\in\mathbb{N}_{+}$, and integer $n\geq f(1)$.

Output: The monochromatic parameter $\mu$ with respect to $y=f(x)$ and $n$.

1

2Let $\mu=0,t=\left\lfloor f^{-1}(n)\right\rfloor$ and $S=\emptyset$;

3 while  _$t\geq 1$_ do

4 if  _$t\notin S$_ then

5 $\mu=\mu+1$;

6 if  _$f^{-1}(t)\geq 1$ is an integer_ then

7 $s=t$;

8 $f^{-1}(s)=f^{-1}(t)$;

9 while  _$f^{-1}(s)\geq 1$ is an integer_ do

10 $S=S\cup\left\\{s,f^{-1}(s)\right\\}$;

11 $\mu=\mu+1$;

12 $s=f^{-1}(s)$;

13

14 end while

15 $t=t-1$;

16

17 else

18 $S=S\cup\\{t\\}$;

19 $t=t-1$;

20

21 end if

22

23 else

24 $t=t-1$;

25

26 end if

27

28 end while

return _$\mu$._

Algorithm 2 Calculate the monochromatic parameter $\mu$ with respect to
$y=f(x)$ and $n$.

######  Remark 1.

In order for Algorithm 2 to run properly, the While-Do loop from step $8$ to
step $12$ cannot be a dead loop, that is, there cannot be a
$x_{0}\in\mathbb{N}_{+}$ such that $f(x_{0})=x_{0}$. In fact, we do not need
to worry about this problem. Since $y=f(x)$ is strictly monotonically
increasing, $f(1)\geq 2$ and $f(x)\in\mathbb{N}_{+}$ for all
$x\in\mathbb{N}_{+}$, it follows that $f(2)\geq 3$. Otherwise, this
contradicts the strictly monotonically increasing of $y=f(x)$. According to
recursion, for all $x\in\mathbb{N}_{+}$, $f(x)\geq x+1$. Therefore, there is
no such $x_{0}\in\mathbb{N}_{+}$ that $f(x_{0})=x_{0}$.

In addition, for binary function equations such that $f(1)=1$, $f(2)\geq 3$,
and strictly monotonically increasing (for example, $y=x^{a}$, where integer
$a\geq 2$), we only need to modify some of the conditions for Algorithm 2,
that is, to replace $t\geq 1$, $f^{-1}(t)\geq 1$ and $f^{-1}(s)\geq 1$ in
steps $2$, $5$ and $8$ with $t\geq 2$, $f^{-1}(t)\geq 2$ and $f^{-1}(s)\geq
2$, respectively.

##  4 Results for Gallai–Rado numbers

We first prove that the Rado number for equation $y=ax+b$ does not exist,
except for $y=x$.

######  Lemma 4.1.

Let integers $k\geq r\geq 2$. If $\mathcal{E}$ is not $r$-regular, then
$\operatorname{R}_{k}(\mathcal{E})$ does not exist.

###### Proof.

Since the equation $\mathcal{E}$ is not $r$-regular, there exists an
$r$-coloring $\chi$ of $\mathbb{N}_{+}$, so that the equation $\mathcal{E}$
has no monochromatic solution in $\mathbb{N}_{+}$. For each integer $k\geq t$,
we assign new colors to some numbers of the same color based on the
$r$-coloring $\chi$ of $\mathbb{N}_{+}$, thereby constructing a $k$-coloring
$\chi^{\prime}$ of $\mathbb{N}_{+}$. Noticing that $\mathbb{N}_{+}$ is an
infinite set, while $k$ is a finite number of colors, so we can always add new
colors to construct a $k$-coloring $\chi^{\prime}$ of $\mathbb{N}_{+}$, which
implies that the equation $\mathcal{E}$ is not $k$-regular. It follows from
Observation 1 that $\operatorname{R}_{k}(\mathcal{E})$ does not exist. ∎

######  Theorem 4.2.

For integers $k\geq 2$, $a\geq 1$, $b\geq 0$ and $(a,b)\neq(1,0)$,
$\operatorname{R}_{k}(y=ax+b)$ does not exist.

###### Proof.

From Lemma 4.1, it is sufficient to show that $y=ax+b$ is not $2$-regular. The
red/blue-coloring of $\mathbb{N}_{+}$ constructed is as follows: For integer
$i\geq 1$, we consider the following set and agree that
$\sum_{j=1}^{0}\frac{b}{a^{j+1}}=0$.

$\mathcal{A}_{i}=\left\\{a^{i}\left(\sum_{j=1}^{i-1}\frac{b}{a^{j+1}}+\frac{1}{a}\right),a^{i}\left(\sum_{j=1}^{i-1}\frac{b}{a^{j+1}}+\frac{1}{a}\right)+1,\ldots,a^{i+1}\left(\sum_{j=1}^{i}\frac{b}{a^{j+1}}+\frac{1}{a}\right)-1\right\\}.$

As can be seen, $\bigcup_{i=1}^{\infty}\mathcal{A}_{i}=\mathbb{N}_{+}$. When
the positive integer $i$ is odd, we color all the numbers in set
$\mathcal{A}_{i}$ red; and when the positive integer $i$ is even, we color all
the numbers in set $\mathcal{A}_{i}$ blue. It is easy to verify that under
this coloring, the equation $y=ax+b$ does not have a monochromatic solution in
$\mathbb{N}_{+}$. The result thus follows. ∎

Obviously, with exact $k$-coloring, $\operatorname{R}_{k}(y=x)=k$ and there
never be any rainbow solution of $y=x$, so the following theorem is
straightforward.

######  Theorem 4.3.

For ingeter $k\geq 2$ and an arbitrary equation $\mathcal{E}$, we have

$\operatorname{GR}_{k}\left(y=x:\mathcal{E}\right)=\operatorname{R}_{k}(\mathcal{E}).$

Let integers $n\geq k\geq b+1$ and $b\geq 1$. According to Corollary 2.5, when
$k\geq b+1$, any $k$-coloring of $[n]$, the equation $y=x+b$ must have a
rainbow solution in $[n]$. The following theorem is directly given.

######  Theorem 4.4.

For ingeters $k\geq b+1$, $b\geq 1$ and an arbitrary equation $\mathcal{E}$,
we have

$\operatorname{GR}_{k}\left(y=x+b:\mathcal{E}\right)=k.$

Next, we provide the result of Gallai–Rado numbers involving monochromatic
general linear equations.

######  Theorem 4.5.

For ingeters $a_{i}\geq 1$ for all $i\in[t]$, $t\geq 1$, $b\geq 2$ and $c\geq
0$, we have

$\operatorname{GR}_{b}\left(y=x+b:y=\sum_{i=1}^{t}a_{i}x_{i}+c\right)=\sum_{i=1}^{t}\left(\lambda_{\min}+(i-1)b\right)a_{i}+c,$

where $\lambda_{\min}=\min\left\\{\lambda:\text{ $\lambda\in[b]$ and
$\left(\sum_{i=1}^{t}a_{i}\lambda+c-\lambda\right)\equiv
0\pmod{b}$}\right\\}$. Moreover, if $\lambda_{\min}$ does not exist, then
$\operatorname{GR}_{b}\left(y=x+b:y=\sum_{i=1}^{t}a_{i}x_{i}+c\right)$ also
does not exist.

###### Proof.

Let $N=\sum_{i=1}^{t}\left(\lambda_{\min}+(i-1)b\right)a_{i}+c$ and
$\mathcal{C}_{(y=x+b,\lambda)}=\left\\{\lambda+(i-1)b:i\in\mathbb{N}_{+}\right\\}\subseteq[N-1]$,
where $\lambda\in[b]$ and $\lambda_{\min}=\min\left\\{\lambda:\text{
$\lambda\in[b]$ and $\left(\sum_{i=1}^{t}a_{i}\lambda+c-\lambda\right)\equiv
0\pmod{b}$}\right\\}$. For the lower bound, we only need to construct a
$b$-colored $[N-1]$ such that there is neither a rainbow solution of $y=x+b$
nor a monochromatic solution of $y=\sum_{i=1}^{t}a_{i}x_{i}+c$ in $[N-1]$.
Since there is no rainbow solution of $y=x+b$ in $[N-1]$, it follows from
Corollary 2.5 that for each $\lambda\in[b]$, the set
$\mathcal{C}_{(y=x+b,\lambda)}$ is monochromatic. Therefore, the colored
structure of $[N-1]$ is uniquely determined without considering the order of
colors. Next, we only need to prove that $[N-1]$ does not contain a
monochromatic solution of $y=\sum_{i=1}^{t}a_{i}x_{i}+c$.

To the contrary, if the equation $y=\sum_{i=1}^{t}a_{i}x_{i}+c$ has a
monochromatic solution in $[N-1]$, then there exists $\lambda_{0}\in[b]$ so
that there are integers $x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{t}$
and $y^{\prime}=\sum_{i=1}^{t}a_{i}x^{\prime}_{i}+c$ are all in
$\mathcal{C}_{(y=x+b,\lambda_{0})}$. Since
$x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{t}$ in
$\mathcal{C}_{(y=x+b,\lambda_{0})}$, it follows that there are positive
ingeters $j_{1},j_{2},\ldots,j_{t}$ such that

$\begin{split}&x^{\prime}_{1}=\lambda_{0}+j_{1}b,\\\
&x^{\prime}_{2}=\lambda_{0}+j_{2}b,\\\ &\qquad\vdots\\\
&x^{\prime}_{t}=\lambda_{0}+j_{t}b.\end{split}$

Therefore,

$\begin{split}y^{\prime}&=\sum_{i=1}^{t}a_{i}x^{\prime}_{i}+c\\\
&=\sum_{i=1}^{t}a_{i}(\lambda_{0}+j_{i}b)+c\\\
&=\sum_{i=1}^{t}a_{i}\lambda_{0}+\left(\sum_{i=1}^{t}a_{i}j_{i}\right)\cdot
b+c\\\ &=\lambda_{0}+\left(\sum_{i=1}^{t}a_{i}j_{i}\right)\cdot
b+\sum_{i=1}^{t}a_{i}\lambda_{0}+c-\lambda_{0}\end{split}$

in $\mathcal{C}_{(y=x+b,\lambda_{0})}$ if and only if
$\left(\sum_{i=1}^{t}a_{i}\lambda_{0}+c-\lambda_{0}\right)\equiv 0\pmod{b}$.
We can see that if such $\lambda_{0}$ does not exist, then the equation
$y=\sum_{i=1}^{t}a_{i}x_{i}+c$ cannot have a monochromatic solution in
$\mathbb{N}_{+}$, which implies that
$\operatorname{GR}_{b}\left(y=x+b:y=\sum_{i=1}^{t}a_{i}x_{i}+c\right)$ does
not exist. If such $\lambda_{0}$ exists, then we choose the smallest one,
$\lambda_{\min}$, to obtain

$\begin{split}&y^{\prime}=\sum_{i=1}^{t}a_{i}x^{\prime}_{i}+c\\\
&\geq\lambda_{\min}a_{1}+\left(\lambda_{\min}+b\right)a_{2}+\cdots+\left(\lambda_{\min}+(t-1)b\right)a_{t}+c\\\
&=\sum_{i=1}^{t}\left(\lambda_{\min}+(i-1)b\right)a_{i}+c=N,\end{split}$

which contradicts with $[N-1]$.

For the upper bound, we consider any $b$-colored $[n]\,(n\geq N)$. If the
equation $y=x+b$ has a rainbow solution in $[n]$, then it is done. Otherwise,
the equation $y=x+b$ does not have a rainbow solution in $[n]$, and in this
case, the colored structure of $[n]$ is uniquely determined. This means that
for each $\lambda\in[b]$, the set
$\mathcal{C}_{(y=x+b,\lambda)}=\left\\{\lambda+(i-1)b:i\in\mathbb{N}_{+}\right\\}\subseteq[n]$
is monochromatic. As discussed above, if the defined $\lambda_{\min}$ does not
exist, then
$\operatorname{GR}_{b}\left(y=x+b:y=\sum_{i=1}^{t}a_{i}x_{i}+c\right)$ also
does not exist. Therefore, we assume that $\lambda_{\min}$ exists. In this
case, the equation $y=\sum_{i=1}^{t}a_{i}x_{i}+c$ has a monochromatic solution
in $\mathcal{C}_{(y=x+b,\lambda_{\min})}$, where one of the monochromatic
solution is

$\begin{split}&x_{1}=\lambda_{\min},\\\ &x_{2}=\lambda_{\min}+b,\\\
&\qquad\vdots\\\ &x_{t}=\lambda_{\min}+(t-1)b,\\\
&y=\sum_{i=1}^{t}a_{i}x_{i}+c=\sum_{i=1}^{t}\left(\lambda_{\min}+(i-1)b\right)a_{i}+c.\end{split}$

The result thus follows. ∎

Regarding monochromatic nonlinear binary function equations, we have the
following results.

######  Theorem 4.6.

Let integers $a\geq 1$, $b\geq 0$ and $c\geq 2$. Then

$\operatorname{GR}_{2}\left(y=x+2:y=ax^{c}+b\right)\left\\{\begin{array}[]{ll}\text{does
not exist},&\text{if $a$ and $b$ are odd;}\\\ =a+b,&\text{if $a$ and $b$ have
different parity;}\\\ =a\cdot 2^{c}+b,&\text{if $a$ and $b$ are even.}\\\
\end{array}\right.$

###### Proof.

We distinguish the following three cases to show this theorem.

######  Case 1.

$a$ and $b$ are odd.

According to Observation 3, we only need to construct a red/blue-coloring of
$\mathbb{N}_{+}$ such that there is neither a rainbow solution of $y=x+2$ nor
a monochromatic solution of $y=ax^{c}+b$. In fact, we only need to color all
odd numbers red and all even numbers blue in $\mathbb{N}_{+}$. Under this
coloring, there is clearly no rainbow solution of $y=x+2$. Let $(x_{0},y_{0})$
be any positive integer solution of $y=ax^{c}+b$. If $x_{0}$ is even, then
$x_{0}^{c}$ is also even, indicating that $ax_{0}^{c}$ is also even. Since $b$
is odd, it follows that $y_{0}=ax_{0}^{c}+b$ is odd, which implies that the
colors of $x_{0}$ and $y_{0}$ are different. If $x_{0}$ is odd, then
$x_{0}^{c}$ is also odd, indicating that $ax_{0}^{c}$ is also odd. Since $b$
is odd, it follows that $y_{0}=ax_{0}^{c}+b$ is even, which implies that the
colors of $x_{0}$ and $y_{0}$ are different. Therefore, there is no
monochromatic solution of $y=ax^{c}+b$ in such red/blue-colored
$\mathbb{N}_{+}$. The result thus follows.

######  Case 2.

$a$ and $b$ have different parity.

For the lower bound, we color all odd numbers red and all even numbers blue in
$[a+b-1]$. Since $y=ax^{c}+b$ has no solution in $[a+b-1]$, it follows that
there is no monochromatic solution of $y=ax^{c}+b$ in $[a+b-1]$. Under this
coloring, there is no rainbow solution of $y=x+2$. Thus,
$\operatorname{GR}_{2}\left(y=x+2:y=ax^{c}+b\right)\geq a+b$.

For the upper bound, we consider any red/blue-coloring of $[a+b]$. To the
contrary, suppose that there is a red/blue-coloring of $[a+b]$ such that there
is neither a rainbow solution of $y=x+2$ nor a monochromatic solution of
$y=ax^{c}+b$. Since there is no rainbow solution of $y=x+2$, it follows from
Corollary 2.5 that without considering the order of colors, the coloring is
uniquely determined. Without loss of generality, we assume that all odd
numbers red and all even numbers blue in $[a+b]$. Noticing that due to the
different parity of $a$ and $b$, $a+b$ is odd. In this case, $(1,a+b)$ is a
monochromatic solution of $y=ax^{c}+b$, which is a contradiction. The result
thus follows.

######  Case 3.

$a$ and $b$ are even.

For the lower bound, we color all odd numbers red and all even numbers blue in
$[a\cdot 2^{c}+b-1]$. Noticing that $y=ax^{c}+b$ has only one solution in
$[a\cdot 2^{c}+b-1]$, which is $(1,a+b)$. But since both $a$ and $b$ are even,
$a+b$ is also even, that is, the solution $(1,a+b)$ is not a monochromatic
solution. Also, under this coloring, there is no rainbow solution of $y=x+2$.
Thus, $\operatorname{GR}_{2}\left(y=x+2:y=ax^{c}+b\right)\geq a\cdot 2^{c}+b$.

For the upper bound, we consider any red/blue-coloring of $[a\cdot 2^{c}+b]$.
To the contrary, suppose that there is a red/blue-coloring of $[a\cdot
2^{c}+b]$ such that there is neither a rainbow solution of $y=x+2$ nor a
monochromatic solution of $y=ax^{c}+b$. Since there is no rainbow solution of
$y=x+2$, it follows from Corollary 2.5 that without considering the order of
colors, the coloring is uniquely determined. Without loss of generality, we
assume that all odd numbers red and all even numbers blue in $[a\cdot
2^{c}+b]$. Since both $a$ and $b$ are even, it follows that $a\cdot 2^{c}+b$
is also even. In this case, $(2,a\cdot 2^{c}+b)$ is a monochromatic solution
of $y=ax^{c}+b$, which is a contradiction. The result thus follows. ∎

It follows from Corollary 2.5 that a $b$-colored $\mathbb{N}_{+}$ contains no
rainbow solution of $y=x+b$ if and only if for each $\lambda\in[b]$, the
$\lambda$-class $\mathcal{C}_{(y=x+b,\lambda)}\subseteq\mathbb{N}_{+}$ is
monochromatic. For example, $\mathcal{C}_{(y=x+2,1)}$ is the set of all
positive odd numbers, and $\mathcal{C}_{(y=x+2,2)}$ is the set of all positive
even numbers. Since $\mathcal{C}_{(y=x+2,1)}$ and $\mathcal{C}_{(y=x+2,2)}$
are arithmetic sequences, it follows that $(x_{0},ax_{0}^{c}+b)$ is a solution
of $y=ax^{c}+b$ in $\mathcal{C}_{(y=x+2,1)}$ or $\mathcal{C}_{(y=x+2,2)}$ if
and only if $\frac{ax_{0}^{c}+b-x_{0}}{2}$ is an integer. If $a$ and $b$ are
odd, then $\frac{ax_{0}^{c}+b-x_{0}}{2}$ is not an integer for any
$x_{0}\in\mathbb{N}_{+}$, which implies that there is no monochromatic
solution of $y=ax^{c}+b$ in $\mathbb{N}_{+}$. If $a$ and $b$ have different
parity, then $\frac{a\cdot 1^{c}+b-1}{2}$ is an integer, which implies that
one of the monochromatic solution of $y=ax^{c}+b$ in $\mathbb{N}_{+}$ is
$(1,a+b)$. If $a$ and $b$ are even, then $\frac{ax_{0}^{c}+b-x_{0}}{2}$ is not
an integer for $x_{0}=1$ but is an integer for $x_{0}=2$, which implies that
one of the monochromatic solution of $y=ax^{c}+b$ in $\mathbb{N}_{+}$ is
$(2,a\cdot 2^{c}+b)$. Based on the above ideas, we provide the following
result.

######  Theorem 4.7.

For a strictly monotonically increasing binary function equation $y=f(x)$ such
that $f(x)\in\mathbb{N}_{+}$ for all $x\in\mathbb{N}_{+}$, and integer $b\geq
2$, we have

$\operatorname{GR}_{b}\left(y=x+b:y=f(x)\right)=f(x_{\min}),$

where $x_{\min}=\min\left\\{x\in\mathbb{N}_{+}:\text{$\frac{f(x)-x}{b}$ is an
integer}\right\\}$. Moreover, if $x_{\min}$ does not exist, then
$\operatorname{GR}_{b}\left(y=x+b:y=f(x)\right)$ also does not exist.

###### Proof.

Let the $\lambda$-class
$\mathcal{C}_{(y=x+b,\lambda)}=\\{\lambda,\lambda+b,\lambda+2b,\ldots\\}\subseteq\mathbb{N}_{+}$
for each $\lambda\in[b]$, and the coloring $\chi$ of $\mathbb{N}_{+}$ is as
described in Corollary 2.5. Specifically, the $\lambda$-class
$\mathcal{C}_{(y=x+b,\lambda)}$ is monochromatic for each $\lambda\in[b]$, and
for different $\lambda_{i}$ and $\lambda_{j}$ in $[b]$,
$\mathcal{C}_{(y=x+b,\lambda_{i})}$ and $\mathcal{C}_{(y=x+b,\lambda_{j})}$
have different colors. Obviously, under the coloring $\chi$, $y=x+b$ does not
have a rainbow solution in $\mathbb{N}_{+}$. Noticing that for each
$\lambda\in[b]$, $\mathcal{C}_{(y=x+b,\lambda)}$ is an arithmetic sequence.
Therefore, $(x_{0},f(x_{0}))$ is a solution of $y=ax^{c}+b$ in
$\mathcal{C}_{(y=x+b,\lambda)}$ for some $\lambda\in[b]$ if and only if
$\frac{f(x_{0})-x_{0}}{b}$ is an integer. Let
$x_{\min}=\min\left\\{x\in\mathbb{N}_{+}:\text{$\frac{f(x)-x}{b}$ is an
integer}\right\\}$. If $x_{\min}$ does not exist, then
$\frac{f(x_{0})-x_{0}}{b}$ is not an integer for any $x_{0}\in\mathbb{N}_{+}$,
which implies that there is no monochromatic solution of $y=ax^{c}+b$ in
$\mathbb{N}_{+}$, and thus $\operatorname{GR}_{b}\left(y=x+b:y=f(x)\right)$
does not exist.

Next, we assume that $x_{\min}$ exists. For the lower bound, we apply the
coloring $\chi$ to $\mathcal{C}_{(y=x+b,\lambda)}\subseteq[f(x_{\min})-1]$. If
there is a monochromatic solution of $y=f(x)$ in $[f(x_{\min})-1]$, then it
contradicts the minimality of $x_{\min}$. Therefore, under the coloring
$\chi$, there is neither a rainbow solution of $y=x+b$ nor a monochromatic
solution of $y=f(x)$ in $[f(x_{\min})-1]$. For the upper bound, we consider
any $b$-coloring of $[f(x_{\min})]$. To the contrary, suppose that there is a
$b$-coloring of $[f(x_{\min})]$ such that there is neither a rainbow solution
of $y=x+b$ nor a monochromatic solution of $y=f(x)$. Since there is no rainbow
solution of $y=x+b$, it follows from Corollary 2.5 that without considering
the order of colors, the coloring $\chi$ is uniquely determined. But under the
coloring $\chi$, we can find a monochromatic solution of $y=f(x)$ in
$[f(x_{\min})]$, which is $(x_{\min},f(x_{\min}))$, a contradiction. The
result thus follows. ∎

##  5 Conclusion

When studying the Gallai–Rado number
$\operatorname{GR}_{k}(\mathcal{E}_{1}:\mathcal{E}_{2})$, it is important that
there is no rainbow solution of the equation $\mathcal{E}_{1}$ in $[n]$ with a
specific colored structure. The rainbow number of the equation
$\mathcal{E}_{1}$ provides some valuable information. We can know with at
least how many colors are needed to color $[n]$, and there always be a rainbow
solution of the equation $\mathcal{E}_{1}$ in $[n]$. For example, it follows
from $\operatorname{rb}([n],y=x+b)=b+1$ that for colored $[n]$ with $b+1$
colors, there always be a rainbow solution of the equation $y=x+b$ in $[n]$,
while with $b$ colors, there may not be a rainbow solution of the equation
$y=x+b$ in $[n]$.

Noticing that the value of $\operatorname{rb}([n],y=x+b)$ dose not rely on
$n$, so no matter how large $n$ is, as long as $[n]$ is colored with $b$
colors, its colored structure without rainbow solution of the equation $y=x+b$
is uniquely determined (see Corollary 2.5). However, for the general linear
binary equation $y=ax+b$, where $a\geq 2,b\geq 0$, the rainbow number
$\operatorname{rb}([n],y=ax+b)$ rely on $n$, which leads to the fact that for
different $n$, the maximum number of colors required to satisfy the equation
$y=ax+b$ without rainbow solution in $[n]$ is also different. It can be seen
that if $\operatorname{rb}([n],\mathcal{E})$ does not rely on $n$, then the
study of the Gallai–Rado number involving rainbow $\mathcal{E}$ becomes much
easier. A natural question is whether there exists a nonlinear binary function
equation $y=f(x)$ whose rainbow number does not rely on $n$?

Since the number of colors of the Gallai–Rado numbers studied in this paper is
fixed at $b$, where the rainbow equation is $y=x+b$, the following problems
can be considered. When the number of colors $k$ satisfies $2\leq k\leq b-1$,
for different equations $\mathcal{E}$, study the Gallai–Rado numbers
$\operatorname{GR}_{k}(y=x+b:\mathcal{E})$. In fact, when $2\leq k\leq b-1$,
the difficulty lies in the fact that the colored structure of $[n]$ that does
not have rainbow solution of the equation $y=x+b$ is not uniquely determined.
On the other hand, we can change the rainbow equation $y=x+b$ studied in this
paper to study other rainbow equations, such as studying
$\operatorname{GR}_{k}(y=ax:\mathcal{E})$ with a given number of colors $k$
and a positive integer $a\geq 2$.

## References

  * [1] H. L. Abbott, D. Hanson, A problem of Schur and its generalizations, Acta Arith. 20 (1972), 175–187.
  * [2] A. Beutelspacher, W. Brestovansky, Generalized Schur Numbers, in: Lecture Notes in Math., Vol. 969, Springer-Verlag, Berlin, 1982, pp. 30–38.
  * [3] M. Budden, Schur numbers involving rainbow colorings, Ars Math. Contem. 18(2) (2020), 281–288.
  * [4] S. Dwivedi, A. Tripathi, On the two-color Rado number for $x_{1}+ax_{2}-x_{3}=c$, INTEGERS 20 (2020), #A36.
  * [5] K. Fallon, C. Giles, H. Rehm, S. Wagner, N. Warnberg, Rainbow numbers of $[n]$ for $\sum_{i=1}^{k-1}x_{i}=x_{k}$, Australas. J. Combin. 77(1) (2020), 1–8.
  * [6] R. J. Faudree, R. Gould, M. Jacobson, C. Magnant, Ramsey numbers in rainbow triangle free colorings, Australas. J. Combin. 46 (2010), 269–284.
  * [7] T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar 18 (1967), 25–66.
  * [8] R. L. Graham, Some of my favorite problems in Ramsey theory, Comb. Number Theory 7(2) (2007), 229–236.
  * [9] H. Harborth, S. Maasberg, Rado numbers for $a(x+y)=bz$, J. Combin. Theory, Ser. A 80 (1997), 356–363.
  * [10] M. Heule, Schur number five, Thirty-Second AAAI Conference on Artificial Intelligence. 2018\.
  * [11] B. Hopkins, D. Schaal, On Rado numbers for $\sum_{i=1}^{m-1}a_{i}x_{i}=x_{m}$, Adv. Appl. Math. 35 (2005), 433–441.
  * [12] I. Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157–162.
  * [13] S. Jones, D. Schaal, Some 2-color Rado numbers, Congr. Numer. 152 (2001), 197–199.
  * [14] W. Kosek, D. Schaal, Rado numbers for the equation $\sum_{i=1}^{m-1}x_{i}+c=x_{m}$, for negative values of $c$, Adv. Appl. Math. 27(4) (2001), 805–815.
  * [15] C. Magnant, P. S. Nowbandegani, Topics in Gallai–Ramsey Theory, Springer, 2020.
  * [16] K. Myers, Rado and Rado-type numbers, High honors thesis, Colgate University, Hamilton, 2007.
  * [17] R. Rado, Studien zur Kombinatorik, Math. Zeit. 36 (1933), 424–480.
  * [18] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.
  * [19] A. Robertson, K. Myers, Some two color, four variable Rado numbers, Adv. Appl. Math. 41 (2008), 214–226.
  * [20] D. Sabo, D. Schaal, J. Tokaz, Disjuntive Rado numbers for $x_{1}+x_{2}+c=x_{3}$, INTEGERS 7 (2007), #A29.
  * [21] D. Saracino, The 2-color rado number of $x_{1}+x_{2}+\cdots+x_{n}=y_{1}+y_{2}+\cdots+y_{k}$, Ars Combin. 129 (2016), 315–321.
  * [22] D. Saracino, The 2-color rado number of $x_{1}+x_{2}+\cdots+x_{m-1}=ax_{m}$, Ars Combin. 113 (2013), 81–95.
  * [23] D. Saracino, The 2-color rado number of $x_{1}+x_{2}+\cdots+x_{m-1}=ax_{m}$, II, Ars Combin. 119 (2015), 193–210.
  * [24] D. Saracin, B. Wynne, The 2-color Rado number of $x+y+kz=3w$, Ars Combin. 90 (2008), 119–127.
  * [25] D. Schaal, On generalized Schur numbers, Congr. Numer. 98 (1993), 178–187.
  * [26] D. Schaal, D. Vestal, Rado numbers for $x_{1}+x_{2}+\cdots+x_{m-1}=2x_{m}$, Congr. Numer. 191 (2008), 105–116.
  * [27] D. Schaal, M. Zinter, Rado numbers for the equation $x_{1}+3x_{2}+c=x_{3}$, Congr. Numer. 211 (2012), 65–71.
  * [28] I. Schur, Über die Kongruenz $x^{m}+y^{m}\equiv z^{m}\pmod{p}$, Jahresber. Deutsch. Mathe. Verein. 25 (1916), 114–117.
  * [29] A. Soifer, Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts. In: A. Soifer, Ramsey Theory Yesterday, Today, and Tomorrow. Progress in Mathematics, vol 285. Birkhäuser, Boston, MA, 2011.
  * [30] G. Yang, Y. Mao, C. He, Z. Wang, Some multivarible Rado numbers, arXiv:2203.04126 [math.CO], 2022.