The Ramsey number of loose paths in 3-uniform hypergraphs

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$

STAR-PATH AND STAR-STRIPE BIPARTITE RAMSEY NUMBERS IN MULTICOLORING Communicated by Gholamreza Omidi

Abstract. For given bipartite graphs G1, G2, . . . , Gt, the bipartite Ramsey number bR (G1, G2, . . . , Gt) is the smallest integer n such that if the edges of the complete bipartite graph Kn,n are partitioned into t disjoint color classes giving t graphs H1, H2, . . . , Ht, then at least one Hi has a subgraph isomorphic to Gi. In this paper, we study the multicolor bipartite Ramsey number bR (G1, G2, . . . , Gt), in the case that G1, G2, . . . , Gt being either stars and stripes or stars and a path