On the cover Ramsey number of Berge hypergraphs

Abstract

For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or Rgraph, if the cardinality of each edge belongs to R. An R-graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection f : E(G) → E(H) such that for every e ∈ E(G), e ⊆ f (e). In this note, we define a new type of Ramsey number, namely the cover Ramsey number, denoted as RR (BG 1 , BG 2 ), as the smallest integer n 0 such that for every covering R-uniform hypergraph H on n ≥ n 0 vertices and every 2-edge-coloring (blue and red) of H , there is either a blue Berge-G 1 or a red Berge-G 2 subhypergraph. We show that for every k ≥ 2, there exists some c k such that for any finite graphs G

On the cover Turán number of Berge hypergraphs

“…Very recently, we [26] defined a new type of Ramsey number, namely the cover Ramsey number, denoted as RR (BG 1 , BG 2 ), as the smallest integer n 0 such that for every covering R-uniform hypergraph H on n ≥ n 0 vertices and every 2-edge-coloring (blue and red) of H, there is either a blue Berge-G 1 or a red Berge-G 2 subhypergraph. We show that for every…”