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.

The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erdős-Rényi process (ER). It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f (c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of ER are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer [1] gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke [10] very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of ER-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n 1−ε , ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(n log log n) guesses. This bound is valid also when only black answer-pegs are used. It improves the O(n log n) bound first proven by Chvátal (Combinatorica 3 (1983), [325][326][327][328][329]. We also show that if both black and white answer-pegs are used, then the O(n log log n) bound holds for up to n 2 log log n colors. These bounds are almost tight as the known lower bound of Ω(n) shows. Unlike for k ≤ n 1−ε , simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal non-adaptive strategy (deterministic or randomized) needs Θ(n log n) guesses.

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G 0 as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r − 1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph G N . In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in G N for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F . In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N 0 (F , r, n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1 − o(1) as n → ∞) avoids creating a copy of F for as long as N = o(N 0 ), and prove that any online strategy will a.a.s. create such a copy once N = ω(N 0 ).

The study of the phase transition of random graph processes, and recently in particular Achlioptas processes, has attracted much attention. Achlioptas, D'Souza and Spencer (Science, 2009) gave strong numerical evidence that a variety of edge-selection rules in Achlioptas processes exhibit a discontinuous phase transition. However, Riordan and Warnke (Science, 2011) recently showed that all these processes have a continuous phase transition.In this work we prove discontinuous phase transitions for three random graph processes: all three start with the empty graph on n vertices and, depending on the process, we connect in every step (i) one vertex chosen randomly from all vertices and one chosen randomly from a restricted set of vertices, (ii) two components chosen randomly from the set of all components, or (iii) a randomly chosen vertex and a randomly chosen component.