On Ramsey numbers of 3-uniform Berge cycles

Abstract

For an arbitrary graph G, a hypergraph H is called Berge-G if there is an injection i:V(G)⟶V(H) and a bijection ψ:E(G)⟶E(H) such that for each e=uv∈E(G), we have {i(u),i(v)}⊆ψ(e). We denote by BrG, the family of r-uniform Berge-G hypergraphs. For families F1,F2,…,Ft of r-uniform hypergraphs, the Ramsey number R(F1,F2,…,Ft) is the minimum integer n such that in every hyperedge coloring of the complete r-uniform hypergraph on n vertices with t colors, there exists i, 1≤i≤t, such that there is a monochromatic copy of a hypergraph in Fi of color i. Recently, the extremal problems of Berge hypergraphs have received considerable attention. In this paper, we focus on Ramsey numbers involving 3-uniform Berge cycles and prove that for n≥4, R(B3Cn,B3Cn,B3C3)=n+1. Moreover, for m≥11 and m≥n≥5, we show that [Formula presented]. This is the first result of Ramsey number for two different families of Berge hypergraphs. © 2024 Elsevier B.V.