Ramsey numbers of Berge-hypergraphs and related structures

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$BK_n$isaBerge-$K_n$hypergraph\mathrm{.}Forhigheruniformity,weshowthat$R^4(BK_t, BK_t) = t+1$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.