论文标题
在3-均匀的Berge周期的Ramsey数字上
On Ramsey numbers of 3-uniform Berge cycles
论文作者
论文摘要
对于任意图$ g $，如果有两次二线$φ$ \ MATHCAL {H} $，则称为Berge- $ G $，如果有两只$φ：E（g）\ LongrightArrow E（\ Mathcal {H}）$，以至于每个$ e \ in E（g）中的每个$ e \ in（g）$，我们有$ e \ e \ subseteq或subseteq或）$。我们用$ \ Mathcal {b}^rg $表示，$ r $ runiform berge- $ g $ hypergraphs的家族。对于家庭$ \ Mathcal {h} _1，\ Mathcal {h} _2，\ ldots，\ Mathcal {h} _t _t $ of $ r $ r $ - 均匀的超级格言，ramsey number $ r（\ Mathcal {h} \ Mathcal {h} _t）$是最小的整数$ n $，因此，在每一个$ t $ hyperedge的色彩中，$ \ nathcal {k} _ {n}^r $都有$ \ nathcal {h} _i $ $ $ $ $ $ $ $ 1 $ 1 $ 1 $ 1的单色单色副本。 最近，许多研究人员研究了Berge Hypergraphs的拉姆西问题。 在本文中，我们专注于涉及$ 3 $ - 统一的Berge Cycles Cycles的Ramsey编号，我们证明了$ n \ geq 4 $，$ r（\ Mathcal {b}^3c_n，\ Mathcal {b}^3c_n，\ Mathcal {b}^3c_n，\ Mathcal {b}^3c_3） $ m \ geq 11 $，我们表明$ r（\ mathcal {b}^3k_m，\ Mathcal {b}^3c_n）= m+\ lfloor \ lfloor \ frac {n -1} {n -1} {2} \ rfloor -1。
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.
