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 A ⊂ [n] contain no k-term arithmetic progression? 3. When does every r-colouring of G(n, p) contain a monochromatic copy of H? 4. How many union-free families are contained in P(n)? 5. What is the volume of the metric polytope? The solutions of these problems (and many others) are based on the same fundamental principle: the objects in question exhibit a certain kind of ‘clustering’, which allows one to count them one cluster at a time, using (in each case) a suitable ‘supersaturation’ theorem. Our plan is as follows: In Lecture 1, we will give a relatively gentle introduction to the method, focused on the example of triangle-free subgraphs of G(n, p); in Lecture 2 we will state the general container lemma, and give several simple but important applications; finally, in Lectures 3 and 4, we will discuss some more advanced applications. Lecture 1: What do you do when the 1st moment blows up? In probabilistic combinatorics one is often faced with the following situation: you want to show that (with high probability) no member of some family of ‘bad’ events occurs, but the expected number of such events is large. Such situations often arise when there is positive correlation between the different bad events in your family, and the effect of such correlations can be difficult to bound. In these lectures we will discuss a recently-discovered method of dealing with certain situations of this type, whose basic idea can be summarised as follows: “Independent sets in many ‘natural’ hypergraphs are ‘clustered’ together.” In this first lecture, we will illustrate this idea with a relatively simple, but important example. Definition 1. The extremal number of a graph H with respect to the Erdős-Rényi random graph G(n, p) is defined to be ex ( G(n, p), H ) := max { e(G) : G ⊂ G(n, p) and H 6⊂ G } . Our aim in this lecture is to prove the following theorem of Frankl and Rödl [32]. Theorem 2. If p 1/ √ n, then ex ( G(n, p),K3 ) = ( 1 4 + o(1) ) pn (1) with high probability as n→∞. 1 To see the lower bound, simply consider the intersection of G(n, p) with a copy of Kn/2,n/2; we will prove the upper bound. One natural first attempt at a proof of Theorem 2 would be to define a random variable Xm := ∣∣{G ⊂ G(n, p) : e(G) = m and K3 6⊂ G}∣∣, and calculate the expected value of Xm. If E[Xm] = o(1), then it follows by Markov’s inequality that ex ( G(n, p),K3 ) 6 m with high probability. However, when we do the calculation we obtain instead E[Xm] > ( ex(n,K3) m ) p = (( 1 + o(1) )epn2 4m )m 1 for all m 6 p ( n 2 ) , so this approach fails. What are we to do? Well, the reason the expected value of Xm blows up is that the triangle-free graphs with m edges are ‘clustered’ together, and this creates strong positive correlations between the events encoding their appearance in G(n, p). If we can understand this clustering, we have a chance of grouping them into a relatively small number of ‘bunches’, and dealing with a whole bunch in a single step. To be more precise, we’d like to prove the following ‘container’ theorem. Theorem 3 (The container theorem for triangle-free graphs). For each n ∈ N, there exists a collection G of graphs on n vertices with the following properties: (a) |G| 6 nO(n3/2). (b) Each G ∈ G contains o(n3) triangles. (c) Each triangle-free graph on n vertices is contained in some G ∈ G. In words, this theorem says that there exists a relatively small collection of graphs G, each of which contains few triangles, with the property that every triangle-free graph is a subgraph of some member of G. To motivate the statement, let’s begin by deducing from it a slightly weaker version of Theorem 2. To do so, we will need the following classical ‘supersaturation’ theorem. Theorem 4 (Supersaturation for triangles). For every ε > 0, there exists δ > 0 such that the following holds. If G is a graph on n vertices with e(G) > ( 1 4 + ε ) n edges, then G has at least δn3 triangles. Theorem 4 is a straightforward consequence of Szemerédi’s regularity lemma, and can also be proved by more elementary means, see the exercises. Note that it follows immediately from this theorem that a graph on n vertices with o(n3) triangles has at most ( 1/4 + o(1) ) n2 edges. Now, suppose that G(n, p) contains a triangle-free graph H with m > (1/4 + ε)pn2 edges. By Theorem 3, there exists a graph G ∈ G such that H ⊂ G, and since G contains o(n3) triangles, it follows that e(G) 6 ( 1/4 + o(1) ) n2. However, by Chernoff’s inequality, the probability that such a graph G contains more than m edges of G(n, p) is at most P ( Bin ( e(G), p ) > m ) 6 e−Ω(pn 2).