We develop a notion of containment for independent sets in hypergraphs. For every r r-
uniform hypergraph GG, we find a relatively small collection\mathcal CC of vertex subsets …

Szemerédi's regularity lemma is an important tool in graph theory which has applications
throughout combinatorics. In this paper we prove an analogue of Szemerédi's regularity …

Let A be a subset of an abelian group G with| G|= n. We say that A is sum-free if there do not
exist x, y, z ε A with x+ y= z. We determine, for any G, the maximal density μ (G) of a sum-free …

In 1918, Heinz Prüfer [120] discovered a fascinating relationship between labelled trees with
n vertices and sequences of length n− 2 made of the elements of the set {1, 2,..., n}. This …

In this survey we describe a recently-developed technique for bounding the number (and
controlling the typical structure) of finite objects with forbidden substructures. This technique …

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+
1− 1) N/2d, where the bound is sharp for a disjoint union of complete d-regular bipartite …

In the parameterized k-clique problem, or k-Clique for short, we are given a graph G and a
parameter k≥ 1. The goal is to decide whether there exist k vertices in G that induce a …

In this short survey article, we present an elementary, yet quite powerful, method of
enumerating independent sets in graphs. This method was first employed more than three …

Many important problems in combinatorics and other related areas can be phrased in the
language of independent sets in hypergraphs. Recently Balogh, Morris, and Samotij J …

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 …