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