Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions
Published online by Cambridge University Press: 03 December 2003
Abstract
For a graph H and an integer n, the Turán number 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, . 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, This is motivated by a conjecture of Erdős that asserts that, for every such H,
For two graphs G and H, the Ramsey number 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, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, 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.
- Type
- Research Article
- Information
- Combinatorics, Probability and Computing , Volume 12 , Issue 5-6: This issue contains volume twelve, parts five and six , November 2003 , pp. 477 - 494
- Copyright
- Copyright © Cambridge University Press 2003
Footnotes
Research supported in part by a State of New Jersey grant, a USA–Israel BSF grant, a grant from the Israel Science Foundation, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.
Research supported in part by a USA–Israel BSF Grant, a grant from the Israel Science Foundation, and a Bergmann Memorial Grant.
Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.
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