The Ramsey number of generalized loose paths in hypergraphs

Let H=(V,E)H=(V,E) be an rr-uniform hypergraph. For each 1≤s≤r−11≤s≤r−1, an ss-path Pnr,s of length nn in HH is a sequence of distinct vertices v1,v2,…,vs+n(r−s)v1,v2,…,vs+n(r−s) such that {v1+i(r−s),…,vs+(i+1)(r−s)}∈E(H){v1+i(r−s),…,vs+(i+1)(r−s)}∈E(H) for each 0≤i≤n−10≤i≤n−1. Recently, the Ramsey number of 1-paths in uniform hypergraphs has received a lot of attention. In this paper, we consider the Ramsey number of r/2r/2-paths for even rr. Namely, we prove the following exact result: R(Pnr,r/2,P3r,r/2)=R(Pnr,r/2,P4r,r/2)=(n+1)r2+1.