A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are 2 (1+o(1))ex(n,H) such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs H.We make a first attempt at addressing this enumeration problem for a general bipartite graph H. We show that an upper bound of 2 O(ex(n,H)) on the number of H-free graphs with n vertices follows merely from a rather natural assumption on the growth rate of n → ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdős and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

Let M be an n × m matrix of independent Rademacher (±1) random variables. It is well known that if $n \leq m$ , then M is of full rank with high probability. We show that this property is resilient to adversarial changes to M. More precisely, if $m \ge n + {n^{1 - \varepsilon /6}}$ , then even after changing the sign of (1 – ε)m/2 entries, M is still of full rank with high probability. Note that this is asymptotically best possible as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [17].

We derive asymptotic formulas for the number of integer partitions with given sums of jth powers of the parts for j belonging to a finite, non-empty set J ⊂ N. The method we use is based on the 'principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.

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In the theory of dense graph limits, a graphon is a symmetric measurable function W : [0, 1] 2 → [0, 1]. Each graphon gives rise naturally to a random graph distribution, denoted G(n, W ), that can be viewed as a generalization of the Erdős-Rényi random graph. Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log n for the clique number of G(n, W ) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n, W ) will be Θ( √ n) almost surely. We also give a family of examples with clique number Θ(n α ) for any α ∈ (0, 1), and some conditions under which the clique number of G(n, W ) will be o( √ n), ω( √ n), or Ω(n α ) for α ∈ (0, 1).