SYMMETRIC AND ASYMMETRIC RAMSEY PROPERTIES IN RANDOM HYPERGRAPHS

A celebrated result of Rodl and Ruci\'nski states that for every graph F, which is not a forest of stars and paths of length 3, and fixed number of colours r >= 2 there exist positive constants c,C such that for p <= cn(-1/m2(F)) the probability that every colouring of the edges of the random graph G(n,p) contains a monochromatic copy of F is o(1) (the '0-statement'), while for p >= Cn(-1/m2(F)) it is 1-o(1) (the '1-statement'). Here m(2)(F) denotes the 2-density of F. On the other hand, the case where F is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in G(n,p). Recently, the natural extension of the 1-statement of this theorem to k-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rodl and Schacht. In particular, they showed an upper bound of order n(-1/mk(F)) for the 1-statement, where m(k)(F) denotes the k-density of F. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of thresholds exists if k >= 4: there are k-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour-class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spohel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.