On Ramsey numbers of 3-uniform Berge cycles

TL;DR

This paper determines the Ramsey numbers for 3-uniform Berge cycles and combines different families of Berge hypergraphs, advancing understanding of hypergraph Ramsey theory.

Contribution

It provides the first known results on Ramsey numbers involving two different families of Berge hypergraphs, specifically for 3-uniform Berge cycles and complete hypergraphs.

Findings

Proved that for n ≥ 4, R(𝔅³Cₙ, 𝔅³Cₙ, 𝔅³C₃) = n + 1.

Established that for m ≥ n ≥ 6 and m ≥ 11, R(𝔅³Kₘ, 𝔅³Cₙ) = m + ⌊(n-1)/2⌋ - 1.

First results on Ramsey numbers for two different families of Berge hypergraphs.

Abstract

For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $\Phi :E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq \Phi (e)$. We denote by $\mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$ is the smallest integer $n$ such that in every $t$-hyperedge coloring of $\mathcal{K}_{n}^r$ there is a monochromatic copy of a hypergraph in $\mathcal{H}_i$ of color $i$, for some $1\leq i\leq t$. Recently, the Ramsey problems of Berge hypergraphs have been studied by many researchers. In this paper, we focus on Ramsey number involving $3$-uniform Berge cycles and we prove that for $n \geq 4$, $ R(\mathcal{B}^3C_n,\mathcal{B}^3C_n,\mathcal{B}^3C_3)=n+1.$ Moreover, for $m \geq n\geq 6$ and $m\geq 11$, we show that $R(\mathcal{B}^3K_m,\mathcal{B}^3C_n)= m+\lfloor \frac{n-1}{2}\rfloor -1.$ This is the first result of Ramsey number for two different families of Berge hypergraphs.