Ramsey Numbers of Berge-Hypergraphs and Related Structures
Ramsey Numbers of Berge-Hypergraphs and Related Structures
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DOI：
10.37236/8892
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发表时间：
2018-08
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通讯作者：
Nika Salia;C. Tompkins;Zhiyu Wang;Oscar Zamora
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作者：
Nika Salia;C. Tompkins;Zhiyu Wang;Oscar Zamora
对于图 $G=(V,E)$，超图 $\mathcal{H}$ 称为 Berge-$G$，用 $BG$ 表示，如果存在注入 $f：E(G) \to E(\mathcal{H})$，使得对于每个 $e \in E(G)$，$e \subseteq f(e)$。令拉姆齐数 $R^r(BG,BG)$ 为最小整数 $n$，使得对于 $n$ 个顶点上的完整 $r$-均匀超图的任何 $2$-边着色，存在单色 Berge-$G$ 子超图。在本文中，我们证明了 Berge 团的 2 色 Ramsey 数是线性的。特别是，我们证明 $R^3(BK_s, BK_t) = s+t-3$ 对于 $s,t \geq 4$ 和 $\max(s,t) \geq 5$，其中 $BK_n$ 是 Berge-$K_n$ 超图。为了获得更高的均匀性，我们表明 $R^4(BK_t, BK_t) = t+1$ 对于 $t\geq 6$ 和 $R^k(BK_t, BK_t)=t$ 对于 $k \geq 5$ 和 $t$ 足够大。我们还研究了迹超图、悬浮超图和扩展超图的拉姆齐数。
For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ 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.