Universal graphs omitting finitely many finite graphs

Abstract If F is a family of graphs, then a graph is F -free, if it contains no induced subgraph isomorphic to an element of F . If F is a finite set of finite graphs, λ is an infinite cardinal, we let CF ( F , λ ) be the minimal number of F -free graphs of size λ such that each F -free graph of size λ embeds into some of them. We show that if 2 λ = λ , then CF ( F , λ ) ≤ c (continuum), there are examples such that CF ( F , λ ) is finite but can be arbitrarily large, and give an example such that CF ( F , λ ) ≥ c for any infinite cardinal λ .