The typical structure of graphs without given excluded subgraphs

Let L be a finite family of graphs. We describe the typical structure of L-free graphs, improving our earlier results [2] on the ErdősFrankl-Rödl theorem [6], by proving our earlier conjecture that, for p = p(L) = minL∈L χ(L)− 1, the structure of almost all L-free graphs is very similar to that of a random subgraph of the Turán graph Tn,p. The “similarity” is measured in terms of graph theoretical parameters of L.