Abstract: For two fixed graphs G and H, the Ramsey number r(G,H) is the smallest integer N such that every red-blue coloring of the edges of the complete graph with N vertices contains a red copy of G or a blue copy of H. This number has been studied for a number of different pairs (G,H) of graphs and one of the earliest results in this field determined its value when both G and H are cycles.
In this talk we will concentrate to the hypergraph case. Let C_t denote the loose cycle of order t, that is the 3-uniform hypergraph with vertices v_1,...,v_t and edges v_1v_2v_3, v_3v_4v_5, v_5v_6v_7,..., v_{t-1}v_{t}v_1. The Ramsey number for C_t is again the smallest N such that every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C_t. We prove that the Ramsey number for C_t is asymptotically equal to 5t/4. We will also discuss some related problems.
This talk represents joint work with P. Haxell, T. Luczak, Y. Peng, A. Rucinski, V. Rödl and M. Simonovits.