Lower bound on the size-Ramsey number of tight paths

Abstract

The size-Ramsey number ˆ R ( k ) ( H ) of a k -uniform hypergraph H is the minimum number of edges in a k -uniform hypergraph G with the property that every ‘2-edge coloring’ of G contains a monochromatic copy of H . For k ≥ 2 and n ∈ N , a k -uniform tight path on n vertices P ( k ) n is defined as a k -uniform hypergraph on n vertices for which there is an ordering of its vertices such that the edges are all sets of k consecutive vertices with respect to this order. We prove a lower bound on the size-Ramsey number of k -uniform tight paths, which is, considered assymptotically in both the uniformity k and the number of vertices n , ˆ R ( k ) ( P ( k ) n ) = Ω (cid:0) log( k ) n (cid:1) .