Symmetric and Asymmetric Ramsey Properties in Random Hypergraphs

Abstract

A celebrated result of Rödl and Ruciński 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 kuniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl 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 threshold 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 Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

An algorithmic framework for obtaining lower bounds for random Ramsey problems

“…In a forthcoming paper [8] we extend Theorem 3 to various classes of -graphs other than cliques. Furthermore we find examples of -graphs F , where the threshold p for G ( ) (n, p) ram − − → k F is neither determined by m (F ) nor by a density m(G) of some obstruction -graph G, but rather exhibits some asymmetric behaviour.…”

“…col ← 0 3 while ∃e 1 , e 2 ∈ E(Ĝ) : e 1 ≡ F e 2 inĜ and e 1 ∩ e 2 = ∅ do 4 color e 1 , e 2 with col 5Ĝ ←Ĝ \ {e 1 , e 2 } and col ← col + 1 6 end 7 while ∃e ∈ E(Ĝ) : e does not belong to an F -copy do8 color e with col 9Ĝ ←Ĝ \ {e} and col ← col + 1 10 end 11 Remove isolated vertices inĜ 12 {B 1 , . .…”

Ramsey properties of randomly perturbed graphs: cliques and cycles

“…Given, for It turns out for a wide collection of pairs of graphs H 1 , H 2 , above the threshold from Conjecture 1.2, G(n, p) is not only (H 1 , H 2 )-Ramsey, but robustly and globally so. While [25] does not give explicit bounds on the error terms, making it a little harder to verify that this strengthening is possible, the more recent containers-based proofs given by Gugelmann, Nenadov, Person,Škorić, Steger and Thomas [20] and Hancock, Staden and Treglown [22] allow for the necessary extensions. In Appendix A.3 we prove Theorem 2.10, implementing the modifications that must be made to the existing proofs.…”

“…Finally, there has been significant interest in Ramsey properties of random hypergraphs (see, for example, [13,19,20]). It would be interesting to obtain analogues of our results in the setting of randomly perturbed hypergraphs.…”

Ramsey games near the critical threshold

“…There has been much work extending this result. We will not attempt an exhaustive survey, but refer the interested reader instead to some of the latest progress on hypergraphs [8], the asymmetric case [11], establishing sharp thresholds [18] and the equivalent problem in settings other than the binomial random graph [5,13]. Our particular concern here will be with the following surprising result of Friedgut, Kohayakawa, Rödl, Ruciński, and Tetali [6] regarding two-round Ramsey games against a random Builder.…”