IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Ramsey numbers of Berge-hypergraphs and related structures

  • Oct. 5, 2018
  • 2:30 p.m.
  • LeConte 317R

Abstract

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a $Berge$-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ 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. This is joint work with Nika Salia, Casey Tompkins and Oscar Zamora.

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