When does the K_4-free process stop?

TL;DR

This paper analyzes the K_4-free process, showing that the resulting maximal graph has bounded maximum degree and near-regular structure, resolving conjectures and providing new bounds in Ramsey theory.

Contribution

It proves the maximum degree bound and near-regularity of the K_4-free process, confirming a conjecture and improving previous bounds, with implications in Ramsey theory.

Findings

Maximum degree in G is at most C n^{3/5} (log n)^{1/5} with high probability.

G has Θ(n^{8/5} (log n)^{1/5}) edges and is nearly regular.

The independence number of G is at least Ω(n^{2/5} (log n)^{4/5} / log log n).

Abstract

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.