"Explosive percolation" is said to occur in an evolving network when a macroscopic connected component emerges in a number of steps that is much smaller than the system size. Recent predictions based on simulations suggested that certain Achlioptas processes (much-studied local modifications of the classical mean-field growth model of Erdős and Rényi) exhibit this phenomenon, undergoing a phase transition that is discontinuous in the scaling limit. We show that, in fact, all Achlioptas processes have continuous phase transitions, although related models in which the number of nodes sampled may grow with the network size can indeed exhibit explosive percolation.

It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability the order of the largest component "jumps" from $o(n)$ to at least $\delta n$ in $o(n)$ steps of the process, a phenomenon known as "explosive percolation." We give a simple proof that this is not the case. Our result applies to all "Achlioptas processes," and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above.Comment: Published in at http://dx.doi.org/10.1214/11-AAP798 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of `almost linear' k-uniform hypergraphs.Comment: 28 pages. To appear in Israel Journal of Mathematic

Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f (X) is a function of independent random variables X = (X1, . . . , Xn). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |fWhile this is easy to check, the main disadvantage is that it considers worst-case changes c k , which often makes the resulting bounds too weak to be useful.In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f (X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations why concentration should hold. Indeed, given an event Γ that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where Γ occurs. The point is that the resulting typical changes c k are often much smaller than the worst case ones.To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdős.

In 1973, Erdős asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g 4 for which some g-element vertex-set contains at least g − 2 triples.)We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed 4, we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6 − o(1))n 2 triples and girth larger than . The process iteratively adds random triples subject to the constraint that the girth remains larger than . Our result is best possible up to the o(1)-term, which is a negative power of n.