  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 $n \to
\infty$, the final graph of the H-free process contains no subgraphs F on $v_F
\leq n^{d}$ vertices with maximum density $\max_{J \subseteq F}\{e_J/v_J\} \geq
c$. This extends and generalizes results of Gerke and Makai for the C_3-free
process.
