On Ramsey numbers of 3-uniform Berge cycles

Abstract

For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $\Phi :E(G)⟶E(\mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq \Phi (e)$. We denote by ${\mathcal{B}}^{r}G$, the family of $r$-uniform Berge-$G$ hypergraphs. For families ${\mathcal{H}}_1,{\mathcal{H}}_2,\dots ,{\mathcal{H}}_t$ of $r$-uniform hypergraphs, the Ramsey number $R({\mathcal{H}}_1,{\mathcal{H}}_2,\dots ,{\mathcal{H}}_t)$ 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 $1\le i\le t$. 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 $n\ge 4$, $R({\mathcal{B}}^3{C}_n,{\mathcal{B}}^3{C}_n,{\mathcal{B}}^3{C}_3)=n+1.$ Moreover, for $m\ge n\ge 6$ and $m\ge 11$, we show that $R({\mathcal{B}}^3{K}_m,{\mathcal{B}}^3{C}_n)=m+\lfloor \frac{n-1}{2}\rfloor -1.$ This is the first result of Ramsey number for two different families of Berge hypergraphs.