Ramsey numbers of Berge-hypergraphs and related structures

Nika Salia ; Casey Tompkins ; Zhiyu Wang ; Oscar Zamora¹

¹Central European University/ Universidad de Costa Rica

The electronic journal of combinatorics, Tome 26 (2019) no. 4

Résumé

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.

Détail

DOI : 10.37236/8892

Classification : 05C55, 05C65, 05D10
Mots-clés : Berge-\(G\) hypergraphs, Ramsey number of trace hypergraphs, Ramsey number of suspension hypergraphs, Ramsey number of expansion hypergraphs