Proof of Erdos-Stone-Simonovits For the lower bound,we consider the Turan graph given by =x(H)-1sets of almost equal size n/r]and [n/r].This has roughly the required number of vertices and it is clear that every subgraph of this graph has chromatic number at most x(H)-1. For the upper bound,note that if H has chromatic number x(H),then,provided t is large enough. it can be embedded in a graph G consisting of x(H)sets A1,A2,...,Ax(H)of size t such that the graph between any two disjoint A;and A,is complete.We simply embed any given colour class into a different A.The theorem now follows from an application of the previous lemma. ▣ 3 Proof of Erd˝os-Stone-Simonovits For the lower bound, we consider the Tur´an graph given by r = χ(H) − 1 sets of almost equal size bn/rc and dn/re. This has roughly the required number of vertices and it is clear that every subgraph of this graph has chromatic number at most χ(H) − 1. For the upper bound, note that if H has chromatic number χ(H), then, provided t is large enough, it can be embedded in a graph G consisting of χ(H) sets A1, A2, . . . , Aχ(H) of size t such that the graph between any two disjoint Ai and Aj is complete. We simply embed any given colour class into a different Ai . The theorem now follows from an application of the previous lemma. ✷ 3