Tura'n Numbers of Bipartite Graphs and Related Ramsey-Type Questions

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 rdegenerate 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) O(n2−1/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) O(n1−c/r). This is motivated by a conjecture of Erdős that asserts that, for every such H , ex(n,H) O(n1−1/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) 2 √ . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, r(G,G) 2 √ m logm for some absolute positive constant c.