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Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

Published online by Cambridge University Press:  03 December 2003

Noga Alon*
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
Michael Krivelevich*
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Benny Sudakov*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08540, USAInstitute for Advanced Study, Princeton, NJ 08540, USA

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
Copyright
Copyright © Cambridge University Press 2003

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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.