On Ramsey numbers of 3-uniform Berge cycles

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 psi : E(G) --> E(H) such that for each e = uv is an element of E(G), we have {i(u), i(v)} subset of psi(e). We denote by BrG, the family of r-uniform Berge -G hypergraphs. For families F-1, F-2, ..., F-t of r-uniform hypergraphs, the Ramsey number R (F-1, F-2, ..., F-t) 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 ((BCn)-C-3,(BCn)-C-3, (BC3)-C-3) = n + 1. Moreover, for m >= 11 and m >= n >= 5, we show that R((BKm)-K-3,(BCn)-C-3) = m + [n-1/2 ] - 1. This is the first result of Ramsey number for two different families of Berge hypergraphs. (c) 2024 Elsevier B.V. All rights reserved.