Dense subgraphs in the H-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 $n \to \infty$, 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.Comment: 7 pages, revised versio

When does the K_4-free process stop?

The K_4-free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K_4. Let G be the random maximal K_4-free graph obtained at the end of the process. We show that for some positive constant C, with high probability as $n \to \infty$, the maximum degree in G is at most $C n^{3/5}\sqrt[5]{\log n}$. This resolves a conjecture of Bohman and Keevash for the K_4-free process and improves on previous bounds obtained by Bollob\'as and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has $\Theta(n^{8/5}\sqrt[5]{\log n})$ edges and is `nearly regular', i.e., every vertex has degree $\Theta(n^{3/5}\sqrt[5]{\log n})$. This answers a question of Erd\H{o}s, Suen and Winkler for the K_4-free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least $\Omega(n^{2/5}(\log n)^{4/5}/\log \log n)$, which matches an upper bound obtained by Bohman up to a factor of $\Theta(\log \log n)$. Our analysis of the K_4-free process also yields a new result in Ramsey theory: for a special case of a well-studied function introduced by Erd\H{o}s and Rogers we slightly improve the best known upper bound.Comment: 39 pages, 3 figures. Minor edits. To appear in Random Structures and Algorithm