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On Ramsey Numbers of Sparse Graphs

Published online by Cambridge University Press:  03 December 2003

Alexander Kostochka  and
B Sudakov
Alexander Kostochka*
Affiliation:
University of Illinois, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk, 630090, Russia
B Sudakov*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08540, USA and Institute for Advanced Study, Princeton, NJ 08540, USA
*
(e-mail: kostochk@math.uiuc.edu)
(e-mail: bsudakov@math.princeton.edu)
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Abstract

The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.

In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours. It is easy to see that any (d,n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order such that neither H nor its complement contains a -common graph.

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. 627 - 641
Copyright
Copyright © Cambridge University Press 2003

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Footnotes

†

Partially supported by NSF grant DMS-0099608 and the Dutch–Russian grant NWO-047-008-006.

‡

Partially supported by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.