In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r 2, there exist constants c, C > 0 such thatThe proof of the 1-statement is based on the recent beautiful hypergraph container theorems by Saxton and Thomason, and Balogh, Morris and Samotij. The proof of the 0-statement is elementary.

A seminal result of Hajnal and Szemerédi states that if a graph G with n vertices has minimum degree δ(G) ≥ (r − 1)n/r for some integer r ≥ 2, then G contains a K r -factor, assuming r divides n. Extremal examples which show optimality of the bound on δ(G) are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Túran theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results:• For any r > ℓ ≥ 2, if G is a graph satisfying δ(G) ≥ r−ℓ r−ℓ+1 n+Ω(n) and α ℓ (G) = o(n), that is, a largest K ℓ -free induced subgraph has at most o(n) vertices, then G contains a K r -factor. This is optimal for ℓ = r − 1 and extends a result of Balogh, Molla, and Sharifzadeh who considered the case r = 3.• If a graph G satisfies δ(G) = Ω(n) and α * r (G) = o(n), that is, every induced K r -free r-partite subgraph of G has at least one vertex class of size o(n), then it contains a K r -factor. A similar statement is proven for a general graph

We show that if the second eigenvalue λ of a d‐regular graph G on n∈3Z vertices is at most εd2/false(nprefixlognfalse), for a small constant ε>0, then G contains a triangle‐factor. The bound on λ is at most an O(logn) factor away from the best possible one: Krivelevich, Sudakov and Szabó, extending a construction of Alon, showed that for every function d=d(n) such that Ω(n2/3)⩽d⩽n and infinitely many n∈N, there exists a d‐regular triangle‐free graph G with Θ(n) vertices and λ=Ω(d2/n).

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.