For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1, BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform hypergraph $\mathcal{H}$ on $n \geq n_0$ vertices and every $2$-edge-coloring (blue and red) of $\mathcal{H}$ , there is either a blue Berge-$G_1$ or a red Berge-$G_2$ subhypergraph. We show that for every $k\geq 2$, there exists some $c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) \leq \hat{R}^{[k]}(BG_1, BG_2) \leq c_k \cdot R(G_1, G_2)^3$. Moreover, we show that for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$ such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $\hat{R}^{[k]}(BG,BG) \leq cn$.

中文翻译：

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Berge 超图的封面 Ramsey 数

对于一组固定的正整数 $R$，如果每条边的基数属于 $R$，我们说 $\mathcal{H}$ 是 $R$-uniform hypergraph 或 $R$-graph。如果 $\mathcal{H}$ 的每个顶点对都包含在某个超边中，则 $R$-graph $\mathcal{H}$ 是 \emph{covering}。对于图 $G=(V,E)$，超图 $\mathcal{H}$ 称为 \textit{Berge}-$G$，记为 $BG$，如果存在注入 $f: E (G) \to E(\mathcal{H})$ 使得对于每个 $e \in E(G)$，$e \subseteq f(e)$。在本笔记中，我们定义了一种新的拉姆齐数，即\emph{cover Ramsey 数}，记为 $\hat{R}^R(BG_1, BG_2)$，作为最小整数 $n_0$，使得对于$n \geq n_0$ 顶点上的每个覆盖 $R$-uniform hypergraph $\mathcal{H}$ 和 $\mathcal{H}$ 的每个 $2$-edge-coloring（蓝色和红色），有一个蓝色的 Berge-$G_1$ 或一个红色的 Berge-$G_2$ 子超图。我们证明，对于每个 $k\geq 2$，都存在一些 $c_k$，使得对于任何有限图 $G_1$ 和 $G_2$，$R(G_1, G_2) \leq \hat{R}^{[k ]}(BG_1, BG_2) \leq c_k \cdot R(G_1, G_2)^3$。此外，我们证明，对于每个正整数 $d$ 和 $k$，存在一个常数 $c = c(d,k)$ 使得如果 $G$ 是在 $n$ 个顶点上的最大度数最大的图$d$，然后 $\hat{R}^{[k]}(BG,BG) \leq cn$。