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 $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ 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 $p\geqslant Cn^{-1/m_{2}(F)}$ 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, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ 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 $k\geqslant 4$: 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 internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position. In this work we study the computational complexity of determining the subset that is generated after n particles have been placed. We develop the first algorithm that provably outperforms the naive step-by-step simulation of all particles. Particularly, our algorithm has a running time of O(n log2 n) and a sublinear space requirement of O(n 1/2 logn), both in expectation and with high probability. In contrast to some speedups proposed for similar models in the physics community, our algorithm samples from the exact distribution. To simulate a single particle fast we have to develop techniques for combining multiple steps of a random walk to large jumps without hitting a forbidden set of grid points. These techniques might be of independent interest for speeding up other problems based on random walks.

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. 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 nonadaptive strategy (deterministic or randomized) needs Θ( n log n ) guesses.

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.

For every integer r⩾2 we call a k-uniform hypergraph H on n vertices r-Ramsey-forcing if every r-edge-coloring of the underlying complete graph Kn contains a monochromatic copy of Kk such that its vertices form an edge in H. In this work we determine the threshold for a random k-uniform hypergraph with n vertices to be r-Ramsey-forcing. This settles an open question from Allen, Böttcher, Hladký, and Piguet [Allen, P., J. Böttcher, J. Hladký and D. Piguet, Turánnical hypergraphs, arXiv:1011.1483v1].

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 - 0(1) as n -> infinity avoids creating a copy of F for as long as N = 0 (N-0), and prove that any online strategy will a.a.s. create such a copy once N = omega(N-0). (C) 2011 Elsevier Inc. All rights reserved.

Our goal is to color the edges of a random graph Gn,m (a graph drawn uniformly at random from all graphs on n vertices with exactly m edges) with a fixed number r of colors such that no color class induces a component of size Ω(n) – a so called ‘giant component’. We prove that for every r⩾2 there exists an analytically computable constant cr∗ for which the following holds: For any ccr∗, with probability 1−o(1) every r-edge-coloring of Gn,rcn contains a monochromatic component on linearly many vertices. In other words, we prove that the property in question has a sharp threshold at m=rcr∗n.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step r new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with r available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible.We prove explicit threshold functions for the duration of this game for an arbitrary number of colors r and a large class of graphs F. This extends earlier work for the case r = 2 by Marciniszyn, Mitsche, and Stojakovic. We also prove a similar threshold result for the vertex-coloring analogue of this game.

We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted, and we wish to maximize the total profit. Since this optimization problem is NP-hard, we focus on approximation algorithms. We obtain fast and simple algorithms for the non-rotational scenario with approximation ratios 9 + ϵ and 8 + ϵ, as well as an algorithm with approximation ratio 7 + ϵ that uses more sophisticated techniques; these are the smallest approximation ratios known for this problem. Furthermore, we show how the used techniques can be adapted to the case where rotation by 90° either around the z -axis or around all axes is permitted, where we obtain algorithms with approximation ratios 6 + ϵ and 5 + ϵ, respectively. Finally our methods yield a 3D generalization of a packability criterion and a strip packing algorithm with absolute approximation ratio 29/4, improving the previously best known result of 45/4.