Exponential lower bound for Berge-Ramsey problems

D\"om\"ot\"or P\'alv\"olgyi

arXiv:1906.04288·math.CO·January 23, 2020·Graphs Comb.

Exponential lower bound for Berge-Ramsey problems

D\"om\"ot\"or P\'alv\"olgyi

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TL;DR

This paper establishes an exponential lower bound for Berge-Ramsey problems, advancing understanding of their computational complexity and combinatorial properties.

Contribution

It provides the first exponential lower bound for Berge-Ramsey problems, highlighting their inherent computational difficulty.

Findings

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Proves exponential lower bound for Berge-Ramsey problems

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Demonstrates complexity growth in Berge-Ramsey configurations

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Advances theoretical understanding of hypergraph Ramsey theory

Abstract

We give an exponential lower bound for Berge-Ramsey problems.

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Exponential lower bound for Berge-Ramsey problems

Dömötör Pálvölgyi 111MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary

Gerbner and Palmer [6], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion. A hypergraph $H$ contains a Berge copy of a graph $G$ , if there are injections $\Psi_{1}:V(G)\to V(H)$ and $\Psi_{2}:E(G)\to E(H)$ such that for every edge $uv\in E(G)$ the containment $\Psi_{1}(u),\Psi_{1}(v)\in\Psi_{2}(uv)$ holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of $G$ . If $|E(H)|=|E(G)|$ , then we say that $H$ is a Berge- $G$ , and we denote such hypergraphs by $\mathcal{B}G$ .

The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors [2, 5, 7]. Denote by $R_{r}(\mathcal{B}G;c)$ the size of the smallest $N$ such that no matter how we $c$ -color the $r$ -edges of $K_{N}^{r}$ , the complete $r$ -uniform hypergraph, we can always find a monochromatic $\mathcal{B}G$ . In [2] $R_{r}(\mathcal{B}K_{n};c)$ was studied for $n=3,4$ . In [5] it was conjectured that $R_{r}(\mathcal{B}K_{n};c)$ is bounded by a polynomial of $n$ (depending on $r$ and $c$ ), and they showed that $R_{r}(\mathcal{B}K_{n};c)=n$ if $r>2c$ and $R_{r}(\mathcal{B}K_{n};c)=n+1$ if $r=2c$ , while $R_{3}(\mathcal{B}K_{n};2)<2n$ (also proved in [7]). In [7] a superlinear lower bound was shown for $r=c=3$ and for every other $r$ for large enough $c$ . This was improved in [4] to $R_{r}(\mathcal{B}K_{n};c)=\Omega(n^{d})$ if $c>(d-1)\binom{r}{2}$ and $R_{r}(\mathcal{B}K_{n};c)=\Omega(n^{1+1/(r-2)}/\log n)$ . We further improve these to disprove the conjecture of [5].

Theorem.

$R_{r}(\mathcal{B}K_{n};c)>(1+\frac{1}{r^{2}})^{n-1}$ * if $c>\binom{r}{2}$ .*

Proof.

It is enough to prove the statement for $c=\binom{r}{2}+1$ . For $r=2$ this reduces to the classical Ramsey’s theorem, so we can assume $r\geq 3$ . We can also suppose $n\geq\binom{r}{2}+1=c$ , or the lower bound becomes trivial. Suppose $N\leq(1+\frac{1}{r^{2}})^{n-1}$ . Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in $K_{N}^{r}$ . Color the $r$ -edges of $K_{N}^{r}$ arbitrarily, respecting the following rule: if $\{u,v\}\subset E$ , then the color of $E$ cannot be the forbidden color of $\{u,v\}$ . Since $c>\binom{r}{2}$ , this leaves at least one choice for each edge. Following the classic proof of the lower bound of the Ramsey’s theorem, now we calculate the probability of having a monochromatic $\mathcal{B}K_{n}$ . The chance of a monochromatic $\mathcal{B}K_{n}$ on a fixed set of $n$ vertices for a fixed color is at most $(\frac{c-1}{c})^{\binom{n}{2}}$ , as the fixed color cannot be the forbidden one on any of the pairs of vertices. Thus the expected number of monochromatic $\mathcal{B}K_{n}$ ’s is at most $c\binom{N}{n}(\frac{c-1}{c})^{\binom{n}{2}}$ . If this quantity is less than $1$ , then we know that a suitable coloring exists. Since $c\leq n\leq n!$ , it is enough to show that $N<(\frac{c}{c-1})^{\frac{n-1}{2}}$ , but this is true using $c=\binom{r}{2}+1$ and $r\geq 3$ . ∎

Remarks and acknowledgment

As was brought to my attention by an anonymous referee, my construction for $r=3$ and $c=4$ is essentially the same as the one used in the proof of Theorem 1(ii) in [3] for a different problem, the $4$ -color Ramsey number of the so-called hedgehog. A hedgehog with body of order $n$ is a $3$ -uniform hypergraph on $n+\binom{n}{2}$ vertices such that $n$ vertices form its body, and any pair of vertices from its body are contained in exactly one hyperedge, whose third vertex is one of the other $\binom{n}{2}$ vertices, a different one for each hypderedge. It is easy to see that such a hypergraph is a Berge copy of $K_{n}$ , and while their result, an exponential lower bound for the $4$ -color Ramsey number of the hedgehog, does not directly imply mine, their construction is such that it also avoids a monochromatic $\mathcal{B}K_{n}$ .

It is an interesting problem to determine how $R_{r}(\mathcal{B}K_{n};c)$ behaves if $c\leq\binom{r}{2}$ . The first open case is $r=c=3$ , just like for hedgehogs.

Bibliography7

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] IF THIS IS HERE, THERE MIGHT BE UNCITED REFERENCES!
2[2] M. Axenovich, A. Gyárfás, A note on Ramsey numbers for Berge-G hypergraphs, ar Xiv:1807.10062.
3[3] D. Conlon, J. Fox, V. Rödl, Hedgehogs are not colour blind, Journal of Combinatorics 8(3) 475–485, 2017.
4[4] D. Gerbner, On Berge-Ramsey problems, ar Xiv:1906.02465.
5[5] D. Gerbner, A. Methuku, G. Omidi, M. Vizer, Ramsey problems for Berge hypergraphs, ar Xiv:1808.10434.
6[6] D. Gerbner, C. Palmer, Extremal Results for Berge Hypergraphs, SIAM Journal on Discrete Mathematics 31(4): 2314–2327, 2017.
7[7] N. Salia, C. Tompkins, Z. Wang, O. Zamora, Ramsey numbers of Berge-hypergraphs, ar Xiv preprint ar Xiv:1808.09863.

TL;DR

Contribution

Findings

Abstract

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Exponential lower bound for Berge-Ramsey problems

Theorem**.**

Proof.

Remarks and acknowledgment

Theorem.