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.

We show that for every k∈N there exists C > 0 such that if pk≥Clog8n/n then asymptotically almost surely the random graph G(n,p) contains the kth power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph G(k)(n,p) for p≥Clog8n/n. The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

In his seminal paper from 1952 Dirac showed that the complete graph on n ≥ 3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊n/2⌋ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on n ≥ 3 vertices with minimum in-and out-degree at least n/2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle.A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in-and out-going) edges touching each vertex. The theorems of Dirac and Ghouila-Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = ω (log n/n) with respect to Hamiltonicity is 1/2 ± o(1). For random directed graphs, Hefetz, Steger and Sudakov (2014+) proved an analogue statement, but only for edge probability p = ω (log n/ √ n). In this paper we significantly improve their result to p = ω log 8 n/n , which is optimal up to the polylogarithmic factor.A Hamilton cycle in a graph or a directed graph is a cycle that passes through all the vertices of the graph exactly once, and a graph is Hamiltonian if it contains a Hamilton cycle. Hamiltonicity is one of the central notions in graph theory, and has been intensively studied by numerous researchers. It is well known that the problem of whether a given graph contains a Hamilton cycle is N P-complete. In fact, Hamiltonicity was one of Karp's 21 N P-complete problems [12].Since one can not hope for a general classification of Hamiltonian graphs, as a consequence of Karp's result, there is a large interest in deriving properties that are sufficient for Hamiltonicity. A classic result by Dirac from 1952 [7] states that every graph on n ≥ 3 vertices with minimum degree at least n/2 is Hamiltonian. This result is tight as the complete bipartite graph with parts of sizes that differ by one, K m,m+1 , is not Hamiltonian. Note that this theorem answers the following question: Starting with the complete graph on n vertices K n , what is the maximal integer ∆ such that for any subgraph H of K n with maximum degree ∆, the graph K n − H obtained by deleting the edges of H from K n is Hamiltonian? This question not only asks for a sufficient condition for a graph to be Hamiltonian, it also asks for a quantification for the "local robustness" of the complete graph with respect to Hamiltonicity.A natural generalization of this question is to replace the complete graph with some arbitrary base graph. Recently, questions of this type have drawn a lot of attention under...

A graph G is said to be ℋ(n,Δ)‐universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any ε>0 and any natural number Δ there exists c>0 such that the random graph G(n, p) is asymptotically almost surely ℋ((1−ε)n,Δ)‐universal for p≥c(logn/n)1/Δ. Bypassing this natural boundary, we show that for Δ≥3 the same conclusion holds when p≫n−1Δ−1log5n. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 380–393, 2017

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability p to a deterministic question of whether there exists a finite graph that forms an obstruction.In the second part of the paper we apply this framework to address and solve various open problems. In particular, we extend the result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs to the case of 2 colors and to hypergraph cliques. As a corollary, this proves a matching lower bound for the result of Friedgut, Rödl and Schacht (2010) and, independently, Conlon and Gowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, in the case of cliques and cycles.