2018/08/29 by Salia, Nika, Tompkins, Casey, Wang, Zhiyu +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1808.09863
For a graph G=(V,E), a hypergraph H is called a Berge-G, denoted by BG, if there exists a bijection f: E(G) → E(H) such that for every e ∈ E(G), e ⊆ f(e). Let the Ramsey number Rr(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 R3(BKs, BKt) = s+t-3 for s,t ≥ 4 and max(s,t) ≥ 5 where BKn is a Berge-Kn hypergraph. For higher uniformity, we show that R4(BKt, BKt) = t+1 for t≥ 6 and Rk(BKt, BKt)=t for k ≥ 5 and t sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.