2022/04/27 by Maherani, Leila, Shahsiah, Maryam
#05C55 #05C65 #05D10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2204.12840
For an arbitrary graph G, a hypergraph H is called Berge-G if there is a bijection Φ:E(G)\longrightarrow E( H) such that for each e∈ E(G), we have e⊆ Φ(e). We denote by BrG, the family of r-uniform Berge-G hypergraphs. For families H1, H2,…, Ht of r-uniform hypergraphs, the Ramsey number R(H1, H2,…, Ht) is the smallest integer n such that in every t-hyperedge coloring of Knr there is a monochromatic copy of a hypergraph in Hi of color i, for some 1≤ i≤ t. Recently, the Ramsey problems of Berge hypergraphs have been studied by many researchers. In this paper, we focus on Ramsey number involving 3-uniform Berge cycles and we prove that for n ≥ 4, R(B3Cn,B3Cn,B3C3)=n+1. Moreover, for m ≥ n≥ 6 and m≥ 11, we show that R(B3Km,B3Cn)= m+\lfloor (n-1)/(2)\rfloor -1. This is the first result of Ramsey number for two different families of Berge hypergraphs.