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Star Versus Two Stripes Ramsey Numbers and a Conjecture of Schelp

Published online by Cambridge University Press:  02 February 2012

ANDRÁS GYÁRFÁS  and
GÁBOR N. SÁRKÖZY
ANDRÁS GYÁRFÁS
Affiliation:
Computer and Automation Research Institute, Hungarian Academy of Sciences, PO Box 63, Budapest, H-1518Hungary (e-mail: gyarfas2@gmail.com)
GÁBOR N. SÁRKÖZY
Affiliation:
Computer and Automation Research Institute, Hungarian Academy of Sciences, PO Box 63, Budapest, H-1518Hungary (e-mail: gyarfas2@gmail.com) Computer Science Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA (e-mail: gsarkozy@cs.wpi.edu)
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Abstract

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that δ(G) > , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.

Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n − 1. This extends R(nK2, nK2) = 3n − 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.

It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 179 - 186
Copyright
Copyright © Cambridge University Press 2012

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