Centre for Discrete and Applicable Mathematics

CDAM Research Report, LSE-CDAM-2007-04

March 2007

The Ramsey Number for Hypergraph Cycles II

P.E. Haxell, T. Luczak, Y. Peng, V. Rödl, A. Rucinski, and J. Skokan

Let C⁽³⁾_n denote the 3-uniform tight cycle, that is the hypergraph with vertices v₁, . . . , v_n and edges v₁v₂v₃, v₂v₃v₄, . . . , v_n-1v_nv₁, v_nv₁v₂. We prove that the smallest integer N = N(n) for which every red-blue coloring 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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		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk