A ramsey type problem concerning vertex colourings

Abstract

Let H → k v G denote the fact that for every function π: V( H) → {1, …, k} there is an induced subgraph G′ of H with G′ ≅ G and V( G′) ⊆ π −1( i) for some i. Folkman has shown that for all graphs G and for all positive integers k such a graph H exists. We examine here f( G, k), the minimum order of a graph H for which H → k v G. We show that for any fixed integer k ≥ 2 there are positive constants C 1 and C 2 such that C 1 n 2 ≤ max{ f( G, k): | V( G)| = n} ≤ C 2 n 2 log 2 n.