Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

• Authors: Noga Alon;Michael Krivelevich;Benny Sudakov
• Venue: Combinatorics, Probability and Computing
• Year: 2003

For a graph $H$ and an integer $n$, the Turán number $\ex(n,H)$ is the maximum possible number of edges in a simple graph on $n$ vertices that contains no copy of $H$. $H$ is $r$-degenerate if every one of its subgraphs contains a vertex of degree at most $r$. We prove that, for any fixed bipartite graph $H$ in which all degrees in one colour class are at most $r$, $\ex(n,H)\,{\leq}\,O(n^{2-1/r})$. This is tight for all values of $r$ and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant $c$ such that, for every fixed bipartite $r$-degenerate graph $H$, $\ex(n,H)\,{\leq}\,O(n^{1-c/r}).$ This is motivated by a conjecture of Erdős that asserts that, for every such $H$, $\ex(n,H)\,{\leq}\,O(n^{1-1/r}).$For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the minimum number $n$ such that, in any colouring of the edges of the complete graph on $n$ vertices by red and blue, there is either a red copy of $G$ or a blue copy of $H$. Erdős conjectured that there is an absolute constant $c$ such that, for any graph $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m}$. Here we prove this conjecture for bipartite graphs $G$, and prove that for general graphs $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m \log m}$ for some absolute positive constant $c$.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.