arXiv 2204.128402022-04-27math.CO

关于3-一致Berge圈的Ramsey数

On Ramsey numbers of 3-uniform Berge cycles

Leila Maherani, Maryam Shahsiah

摘要：

对于任意图$G$，如果存在一个双射$Φ:E(G)\longrightarrow E( \mathcal{H})$使得对于每个$e\in E(G)$，我们有$e\subseteq Φ(e)$，则称超图$\mathcal{H}$为Berge- $G$.我们用$\mathcal{B}^rG$表示$r$ -一致Berge- $G$超图族.对于$r$ -一致超图族$\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$，Ramsey数$R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$是最小的整数$n$，使得在$\mathcal{K}_{n}^r$的每个$t$ -超边染色中，对于某些$1\leq i\leq t$，都有一个颜色为$i$的超图在$\mathcal{H}_i$中的单色副本.近年来，Berge超图的Ramsey问题得到了许多学者的研究.本文主要讨论了涉及$3$ -一致Berge圈的Ramsey数，证明了对于$n \geq 4$，$ R(\mathcal{B}^3C_n,\mathcal{B}^3C_n,\mathcal{B}^3C_3)=n+1.$，并且证明了对于$m \geq n\geq 6$和$m\geq 11$，$R(\mathcal{B}^3K_m,\mathcal{B}^3C_n)= m+\lfloor \frac{n-1}{2}\rfloor -1.$，这是关于两类不同Berge超图族的Ramsey数的第一个结果.

英文摘要：

For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $Φ:E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq Φ(e)$. 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.

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英文摘要：

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