Dense subgraphs in the -free process

The H -free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of H is created, where H is some fixed graph. When H is strictly 2 -balanced, we show that for some c , d > 0 , with high probability as n → ∞ , the final graph of the H -free process contains no subgraphs F on v F ≤ n d vertices with maximum density max J ⊆ F { e J / v J } ≥ c . This extends and generalizes results of Gerke and Makai for the C 3 -free process.