Dense subgraphs in the HH-free process

The HH-free process starts with the empty graph on nn vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of HH is created, where HH is some fixed graph. When HH is strictly 22-balanced, we show that for some c,d>0c,d>0, with high probability as n→∞n→∞, the final graph of the HH-free process contains no subgraphs FF on vF≤ndvF≤nd vertices with maximum density maxJ⊆F{eJ/vJ}≥cmaxJ⊆F{eJ/vJ}≥c. This extends and generalizes results of Gerke and Makai for the C3C3-free process.