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The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Published online by Cambridge University Press:  01 March 2009

P. E. HAXELL ,
T. ŁUCZAK ,
Y. PENG ,
V. RÖDL ,
A. RUCIŃSKI  and
J. SKOKAN
P. E. HAXELL
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1 (e-mail: pehaxell@math.uwaterloo.ca)
T. ŁUCZAK
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: tomasz@amu.edu.pl, andrzej@mathcs.emory.edu)
Y. PENG
Affiliation:
Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA (e-mail: mapeng@isugw.indstate.edu)
V. RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30032, USA (e-mail: rodl@mathcs.emory.edu)
A. RUCIŃSKI
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: tomasz@amu.edu.pl, andrzej@mathcs.emory.edu)
J. SKOKAN
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: jozef@member.ams.org)
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Abstract

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

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