Off‐Diagonal Ramsey Numbers for Slowly Growing Hypergraphs

Abstract

ABSTRACT For a k ‐uniform hypergraph and a positive integer , the Ramsey number denotes the minimum such that every ‐vertex ‐free ‐uniform hypergraph contains an independent set of vertices. A hypergraph is slowly growing if there is an ordering of its edges such that for each . We prove that if is fixed and is any non‐ k ‐partite slowly growing ‐uniform hypergraph, then for , In particular, we deduce that the off‐diagonal Ramsey number is of order , where is the triple system . This is the only 3‐uniform Berge triangle for which the polynomial power of its off‐diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs and hypergraph containers.