Ramsey numbers of Berge-hypergraphs and related structures

TL;DR

This paper establishes that the 2-color Ramsey numbers for Berge cliques in hypergraphs grow linearly with the size of the clique, providing exact values for various uniformities and exploring related hypergraph structures.

Contribution

It proves the linearity of the 2-color Ramsey number for Berge cliques and determines exact values for specific uniformities and sizes, advancing understanding of hypergraph Ramsey theory.

Findings

The 2-color Ramsey number of Berge cliques is linear.

Exact values of Ramsey numbers for Berge-$K_s$ in 3-uniform hypergraphs.

Ramsey numbers for higher uniformity Berge cliques are determined.

Abstract

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.