Extremal problems in random graphs

The theory of random graphs was founded by Paul Erdȍs and Alfréd Rényi around 1959. Since then this interesting and fruitful branch of combinatorics attracted many experts from mathematics and theoretical computer science. This book discusses several questions from the realm of classical graph theory in the context of random graphs. In particular, we address so-called Ramsey and Turán type properties of graphs, which are central to the relatively young field of extremal graph theory. Amongst other results, this book establishes an embedding lemma for sparse graphs, which often constitutes the companion to the sparse version of Szemerédi’s regularity lemma. A stronger form of this embedding lemma was conjectured by Kohayakawa, Łuczak, and Rödl in 1994. This book also continues with the work of Kohayakawa and Kreuter from 1997. We prove strong lower bounds on the edge probability of random graphs that typically allow for an edge coloring without certain monochromatic substructures. Supposing the embedding conjecture of Kohayakawa, Łuczak, and Rödl holds, these bounds are tight and give rise to threshold functions.