Triangle-Free Subgraphs of Random Graphs

Abstract

The Andrásfai–Erdős–Sós Theorem [Andrásfai, B., P. Erdős and V. T. Sós, On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Math. 8 (1974), pp. 205–218] states that all triangle-free graphs on n vertices with minimum degree strictly greater than 2n/5 are bipartite. Thomassen [Thomassen, C., On the chromatic number of triangle-free graphs of large minimum degree, Combinatorica 22 (2002), pp. 591–596] proved that when the minimum degree condition is relaxed to (13+ε)n, the result is still guaranteed to be rε-partite, where rε does not depend on n. We prove best possible random graph analogues of these theorems.