The Ramsey number of generalized loose paths in hypergraphs

Let H = ( V , E ) be an r -uniform hypergraph. For each 1 ? s ? r - 1 , an s -path P n r , s of length n in H is a sequence of distinct vertices v 1 , v 2 , ? , v s + n ( r - s ) such that { v 1 + i ( r - s ) , ? , v s + ( i + 1 ) ( r - s ) } ? E ( H ) for each 0 ? 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 / 2 -paths for even r . Namely, we prove the following exact result: R ( P n r , r / 2 , P 3 r , r / 2 ) = R ( P n r , r / 2 , P 4 r , r / 2 ) = ( n + 1 ) r 2 + 1 .