 Enumeration of subgraph-closed graph properties

Here $r+1$ is the least chromatic number of an excluded graph, for a proper unbounded subgraph-closed property. All subgraphs of a balanced Turan graph give the lower bound. For the upper bound, a regularity reduced graph is clique-free, so its dense pairs have the Turan theorem edge bound; sparse pairs contribute only small binary entropy and the bounded partition description costs only a linear number of bits.