Hypergraph containers

The Method of Hypergraph Containers

In this series of four lectures, we will give a fast-paced introduction to a recentlydeveloped technique in probabilistic combinatorics, known as the method of hypergraph containers. This general technique can be applied in a wide range of settings; to give just a few examples, it has been used to study the following questions: 1. What is the largest H-free subgraph of G(n, p)? 2. How many sets...

The Method of Hypergraph Containers

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 exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of ‘containe...

Hypergraph Rewriting

In their pioneering paper [KB70], Knuth and Bendix showed that con uence (or, equivalently, the Church-Rosser property) is decidable for terminating term rewriting systems. It su ces to compute all critical pairs t s ! u of rewrite steps in which s is the superposition of the left-hand sides of two rules, and to check whether t and u reduce to a common term. This procedure is justi ed by the so...

Hypergraph Codes

Hypergraph Games

The above is called the “strong” version of the game. In the “weak” version, also called “maker-breaker”, the second player’s aim is not to occupy a winning set but just to prevent the first player from doing so. The interest is both for general theorems about games and also in particular games of interest, like the Hales-Jewett game (multi-dimensional noughts & crosses). Roughly speaking, a fa...