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Book Ramsey Numbers and Quasi-Randomness

Published online by Cambridge University Press:  11 October 2005

V. NIKIFOROV
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, TN 38152, USA (e-mail: vnikifrv@memphis.edu, ccrousse@memphis.edu, rschelp@memphis.edu)
C. C. ROUSSEAU
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, TN 38152, USA (e-mail: vnikifrv@memphis.edu, ccrousse@memphis.edu, rschelp@memphis.edu)
R. H. SCHELP
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, TN 38152, USA (e-mail: vnikifrv@memphis.edu, ccrousse@memphis.edu, rschelp@memphis.edu)

Abstract

A set of n triangles sharing a common edge is called a book with n pages and is denoted by $B_{n}$. It is known that the Ramsey number $r ( B_{n} ) $ satisfies $r ( B_{n} ) = ( 4+o ( 1 ) ) n.$ We show that every red–blue edge colouring of $K_{ \lfloor ( 4-\varepsilon ) n \rfloor }$ with no monochromatic $B_{n}$ exhibits quasi-random properties when $\varepsilon$ tends to 0. This implies that there is a constant $c>0$ such that for every red–blue edge colouring of $K_{r ( B_{n} ) }$ there is a monochromatic $B_{n}$ whose vertices span at least $ \lfloor cn^{2} \rfloor $ edges of the same colour as the book.

As an application we find the Ramsey number for a class of graphs.

Type
Paper
Copyright
2005 Cambridge University Press

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