On the Erd{\H o}s--Hajnal problem in the case of 3-graphs

TL;DR

This paper investigates the minimal edges in 3-uniform hypergraphs that are not r-colorable, compares existing methods, and improves the lower bounds for this Erdős–Hajnal problem.

Contribution

It provides a comparative analysis of methods and advances the lower bounds for the minimal edges in non-r-colorable 3-uniform hypergraphs.

Findings

Comparison of existing methods for 3-uniform hypergraphs

Improved lower bounds for m(3,r)

Enhanced understanding of the Erdős–Hajnal problem in 3-graphs

Abstract

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit. The only trivial case is $n=2$ in which $m(2,r) = \binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.

Equations

$$\frac{m(n,r)}{r^n}$$

$$m(n,r)=\left(\genfrac{}{}{0ex}{}{(n-1)r+1}{n}\right),$$

$$m(n,r)<\underset{a\ge 0}{\min }T(r(n+a-1)+1,n+a,n),$$

$$m(n,r)<C{n}^3\ln n\cdot r^n.$$

$$L_3\le \frac{4}{3}.$$

$$L_n=\frac{(n-1{)}^n}{n!}.$$

$$|E|\cdot a^{1-n}.$$

$$|E|<a^{n-1}(r-a)(n-1)$$

$$m(n,r)\ge (n-1)\lceil \frac{r}{n}\rceil {\lfloor \frac{n-1}{n}r\rfloor }^{n-1}.$$

$$\frac{[(n-1)!{]}^2[(n-2)!{]}^{r-2}}{((n-1)r+1)!}.$$

$$2\frac{|E{|}^r}{r!}\frac{[(n-1)!{]}^2[(n-2)!{]}^{r-2}}{((n-1)r+1)!}<1,$$

$$m(n,r)>c\sqrt{n}{r}^n,$$

$$m(n,r)\ge c\frac{n}{\ln n}{r}^n,$$

$$m(n,r)=\min \{N:f(N)>r\}.$$

$$f(N)⩽\underset{a_1+a_2+\cdots +a_p⩽N/p^{n-1}}{\max }f(a_1)+f(a_2)+\cdots +f(a_p).$$

$$f(N)⩽N^{1/n}\cdot \underset{M⩽a<c_{n}M}{\max }f(a)\cdot a^{-1/n}$$

$$L_3>\mathrm{0.324.}$$

$$L_3\ge \frac{4}{e^2}>\mathrm{0.54.}$$

$$\frac{|E|}{2}{\left(\frac{|E|}{r-1}\right)}^{r-1}$$

$$I(G,P_r)\le \frac{n}{2}{\left(\frac{n}{r-1}\right)}^{r-1}.$$

$$\sum _{i=1}^k\frac{n^r}{(r+1{)}^{i(r-1)}}.$$

$$X(q,l)\le \underset{1\le m\le l}{\max }mX(q-1,l-m-1)\le \underset{1\le m\le l}{\max }mX(q-1,l-m).$$

$$X(q,l)\le {\left(\frac{l}{q}\right)}^q.$$

$$X(q,l)\le \underset{1\le m\le l}{\max }m{\left(\frac{l-m}{q-1}\right)}^{q-1}.$$

$$\frac{[(n-1)!{]}^2[(n-2)!{]}^{r-2}}{((n-1)r+1)!}=\frac{4}{(2r+1)!}.$$

$$\frac{|E{|}^r}{2(r-1{)}^{r-1}}\frac{4}{(2r+1)!}<1,$$

$$L_3\ge \underset{r\to \infty }{\lim }\sqrt[r]{\frac{(2r+1)!(r-1{)}^{r-1}}{2}}\frac{1}{r^3}=\frac{4}{e^2}.$$

$$\underset{v\in e}{\max }w(v)-\underset{v\in e}{\min }w(v)\le \frac{1-p}{a};$$

$$P\left[e\text{mbox}isbad\right]={\left(\frac{1-p}{a}\right)}^{n-1}\left(\frac{1-p}{a}+n\left(1-\frac{1-p}{a}\right)\right)\le n{\left(\frac{1-p}{a}\right)}^{n-1}=3{\left(\frac{1-p}{a}\right)}^2.$$

$$P\left[C(A_1,\dots ,A_a)\right]\le a^{-a(n-2)}\frac{p^{a-1}}{(a-1)!}=a^{-a}\frac{p^{a-1}}{(a-1)!}.$$

Full text

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has a limit.

The only trivial case is $n=2$ in which $m(2,r)=\binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.

1 Introduction

A hypergraph $H=(V,E)$ consists of a finite set of vertices $V$ and a family $E$ of the subsets of $V$, which are called edges. A hypergraph is called $n$-uniform if every edge has size $n$. A vertex $r$-coloring of a hypergraph $H=(V,E)$ is a map from $V$ to $\{1,\ldots,r\}$. A coloring is proper if there is no monochromatic edges, i.e., any edge $e\in E$ contains two vertices of different color. The chromatic number of a hypergraph $H$ is the smallest number $\chi(H)$ such that there exists a proper $\chi(H)$-coloring of $H$. Let $m(n,r)$ be the minimal number of edges in an $n$-uniform hypergraph with chromatic number more than $r$.

We are interested in the case when $n$ is much smaller than $r$ (see [9] for general case and related problems). Erdős and Hajnal [6] introduced problems on determining $m(n,r)$ and related quantitites.

1.1 Upper bounds

Erdős conjectured [5] that

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for $r>r_{0}(n)$, that is achieved on the complete hypergraph.

However Alon [2] disproved the conjecture by using the estimate

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where the Turán number $T(v,k,n)$ is the smallest number of edges in an $n$-uniform hypergraph on $v$ vertices such that every induced subgraph on $k$ vertices contains an edge. Different bound on Turán numbers give better refine the complete hypergraph construction when $n>3$ (see [10] for a survey). So the case $n=3$ is in some sense the most interest.

Also note, that using the same inequality with better bounds on Turán numbers Akolzin and Shabanov [1] showed that

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Alon conjectured that the sequence $m(n,r)/r^{n}$ has a limit which was proved by Cherkashin and Petrov [3]. Denote the corresponding limit be $L_{n}$. In this paper we are interested in estimates on $L_{3}$. The best known upper bound follows from the complete hypergraph:

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1.2 Lower bounds

There are several ways to show an inequality of type $m(n,r)\geq c(n)r^{n}$ (i.e. $L_{n}\geq c(n)$). Note that Erdős–Hajnal conjecture implies in particular that

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Alon [2] suggested to color vertices of an $n$-uniform hypergraph in $a<r$ colors uniformly and independently, and then recolor a vertex in every monochromatic edge in unused color. The expected number of monochromatic edges is

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Note that we have $r-a$ remaining colors, and we can color $n-1$ vertices in each unused color such that no new monochromatic edge appears. Summing up, if

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then a hypergraph $H=(V,E)$ has a proper $r$-coloring. Substituting $a=\left\lfloor\frac{n-1}{n}r\right\rfloor$, we get

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This method gives $L_{3}\geq 8/27>0.296$.

Another way is due to Pluhár [8]. He introduced the following useful notion. A sequence of edges $a_{1},\ldots,a_{r}$ is an $r$-chain if $|a_{i}\cap a_{j}|=1$ if $|i-j|=1$ and $a_{i}\cap a_{j}=\emptyset$ otherwise; it is an ordered $r$-chain if $i<j$ implies that every vertex of $a_{i}$ is not bigger than any vertex of $a_{j}$ (with respect to a certain fixed linear ordering on $V$).

Pluhár’s theorem states that existence of an order on $V$ without ordered $r$-chains is equivalent to $r$-colorability of $H=(V,E)$. Let us prove a lower bound on $m(n,r)$ via this theorem. Consider a random order on the vertex set. Note that the probability of an $r$-chain to be ordered is

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From the other hand, the number of $r$-chains is at most $2|E|^{r}/r!$ since every set of $r$ edges generates at most 2 chains. So if

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then we have a proper $r$-coloring of $H$. After taking $r$-root and some calculations we have

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and in particular $L_{3}\geq 4/e^{3}>0.199$.

Combining two previous arguments with Cherkashin–Kozik approach [4] Akolzin and Shabanov [1] proved that

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without explicit bounds on $c$. We show that this method gives the bound $L_{3}>0.205$ in Section 3.

Cherkashin and Petrov [3] suggested an approach, based on the evaluation of the inverse function, to show that the sequence $m(n,r)/r^{n}$ has a limit. Denote by $f(N)$ the maximal possible chromatic number of an $n$-uniform hypergraph with $N$ edges. Also $f(0)=1$ by agreement. The function $f$ non-strictly increases and satisfies

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Therefore $m(n,r)\sim Cr^{n}$ if and only if $f(N)\sim(N/C)^{1/n}$. The following lemmas were proved in [3].

Lemma 1**.**

For any $N>0$ and any positive integer $p$ we have

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Lemma 2**.**

Denote $c_{n}=(1-2^{1-n})^{-n}$. For any $M>0$ the inequality

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holds for all $N\geqslant M$.

It is known that $f(0)=1$, $f(1)=\ldots=f(6)=2$, $f(7)=\ldots=f(26)=3$ (see [akolzin20183]). Lemmas 1, 2 and computer calculations was used to get

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One more way to get a (very weak) bound $L_{n}\geq n^{-n}$ appeared in the same paper [3]. It is based on the straightforward induction on $r$: it was shown that there is a large independent set, so we can color it in color $r$ and apply the inductive assumption.

The contribution of the paper is the following theorem, which is proved by refining Pluhar approach via inducibility arguments.

Theorem 1**.**

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Structure of the paper.

In Section 2 we show how to apply inducibility to the chain argument and proof Theorem 1. In Section 3 we find the constant in Akolzin–Shabanov theorem for $n=3$ and show that even if we apply Theorem 2 to the corresponding part of the proof, the constant will be still worse than in Theorem 1.

2 Inducibility tool

Theorem 2**.**

Suppose $H=(V,E)$ is a hypergraph. Then it has at most

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$r$-chains.

We need a notion of inducibility. Denote by $I(G,H)$ the number of induced subgraphs in $G$, isomorphic to $H$. The following simple bound was proved by Pippenger and Golumbic.

Lemma 3** (Pippenger–Golumbic [7]).**

Let $G$ be a graph on $n$ vertices. Then

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It turns out that the bound is close to optimal.

Example 1**.**

Let $n$ be the $k$-th power of $r+1$, $r>2$. Consider the blow-up of $C_{r+1}$ where the same construction is placed inside the blow-up of each vertex. Every $r$-chain has vertices only in the different copies. Hence the number of chains is

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Proof of Lemma 3.

Let $X(q,l)$ denote the largest possible number of ways of sequentially choosing $q$ objects $w_{0},w_{1},\dots,w_{q-1}$ from among $l$ objects, subject to rules whereby the set of objects that are eligible to be chosen as $w_{i}$ depends only on the previous choices $w_{0},w_{1},\dots,w_{i-1}$, and whereby no object that is eligible to be chosen as $w_{i}$ will be eligible to be chosen as $w_{j}$ for any $i+1\leq j\leq q-1$. Clearly, $X(0,l)=1$. If $q>0$, let $m$ denote the number of objects eligible to be chosen as $w_{0}$. For any choice of $w_{0}$, the remaining $q-1$ objects can be chosen in at most $X(q-1,l-m)$ ways. Thus

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From these relations, we obtain

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by induction on $q$: the base $q=1$ is obvious. To prove the step it is enough to maximize the right-hand side of

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Taking derivative, we get the maximum at $m=l/q$, and we are done.

Also there are $n$ ways to choose the first vertex. Obviously, it is an upper bound on the number of induced $r$-paths, multiplied by 2, because every path is counted twice. ∎

Proof of Theorem 2.

Consider an auxiliary graph $G=(E,F)$ with vertex set is the edge set of $H$ and edges connect pairs of graph vertices, which intersect (as hyperedges) on exactly one vertex. The number of $r$-chains is at most the number of induced graphs $P_{r}$ in $G$, because every $r$-chain forms induced $P_{r}$ (note that the reverse consequence is wrong, because non-edge in $G$ can mean that the corresponding hyperedges have large intersection, which is impossible in $r$-chain). Hence, Lemma 3 finishes the proof. ∎

Proof of Theorem 1.

Let us try to color by the Pluhar greedy algorithm. Recall that the probability of an $r$-chain to be ordered is

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Using Theorem 2 we get that if

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than hypergraph is $r$-colorable. Summing up,

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∎

3 Analysis of the Akolzin–Shabanov proof

We rewrite the proof from [1] to get optimal constant in the case $n=3$.

First, for every vertex $v$ introduce the weight $w(v)$ as randomly (accordingly to the uniform distribution and independently) chosen number from $[0,1]$. Fix parameters $p\in[0,1]$, $a<r$. An edge $e$ is called bad if

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otherwise it is called good.

The coloring algorithm is the following. First we color a (random) subhypergraph, consisting of all good edges, in $a$ colors via Pluhar approach; then we color (or recolor) some vertices from bad edges in unused $r-a$ colors. If Pluhar approach succeed (i.e. there is no ordered $a$-chains) and we have at most $(n-1)(r-a)$ bad edges, then the algorithm return a proper $r$-coloring. Let us evaluate the probability of success.

Lemma 4** (Akolzin–Shabanov [1]).**

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Let $C(A_{1},\dots,A_{a})$ denote the event that all the edges $A_{j}$ are good and $(A_{1},\dots,A_{a})$ is an ordered $a$-chain.

Lemma 5** (Akolzin–Shabanov [1]).**

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By Theorem 2 we have at most $(|E|/a)^{a}$ $a$-chains. Define $c=|E|/r^{3}$; we need

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Also we need at most $(n-1)(r-a)=2(r-a)$ bad edges:

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Define $x=r/a$. Then we need $cx^{3}pe<1$ and

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Computer simulations gives that for $p=0.741$ and $x=1.05$ the algorithm with $c=0.42$ returns a proper coloring with positive probability, which implies $L_{3}>0.42$.

If we simply follow the initial proof, the required inequalities are

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So pure Akolzin–Shabanov approach gives $L_{3}>0.205$. Both constants are worse than in Theorem 1.

4 Open problems

• •

First, recall that the Erdős conjecture is still open in the case $n=3$.

• •

Also it is natural to ask if $m(n,r)$ is regular on the first variable, i.e.

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• •

In the proof of Theorem 2 we consider an auxiliary graph $G$. The problem is to describe the set of graphs, which may be achieved from an $r$-chromatic $n$-uniform hypergraph.

Acknowledgements.

I am grateful to Alexander Sidorenko for an introduction in inducibility theory and to Fedor Petrov, who noted some deficiencies in the paper. Also Alexander Sidorenko pointed to an inattention, relating to the use of Turán numbers.