Global hypercontractivity and its applications

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556

Turán problems in graphs and hypergraphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. In Chapter 2, we prove an analogue of this result for 3-graphs (3-uniform hy¬pergraphs) together with an associated stability result. Let K− 4 , F5 and F6 be 3-graphs with vertex sets {1, 2,3, 4}, {1, 2,3,4, 5} and {1, 2,3,4, 5, 6} re¬spectively and edge sets E(K−4 ) = {123, 124, 134}, E(F5) = {123, 124, 345}, E(F6) = {123, 124,345, 156} and F = {K4, F6}. For n =6 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tri¬partite 3-graph S3(n). This extends an old result of Bollobas that S3(n) is the unique 3-graph of maximum size with no copy of K− 4 or F5. In 1941, Turán generalised Mantel's theorem to cliques of arbitrary size and then asked whether similar results could be obtained for cliques on hyper-graphs. This has become one of the central unsolved problems in the field of extremal combinatorics. In Chapter 3, we prove that the Turán density of K(3) 5 together with six other induced subgraphs is 3/4. This is analogous to a similar result obtained for K(3) 4 by Razborov. In Chapter 4, we consider various generalisations of the Turán density. For example, we prove that, if the density in C of ¯P3 is x and C is K3-free, then |E(C)| /(n ) ≤ 1/4+(1/4)J1 − (8/3)x. This is motivated by the observation 2 that the extremal graph for K3 is ¯P3-free, so that the upper bound is a natural extension of a stability result for K3. The question how many edges can be deleted from a blow-up of H before it is H-free subject to the constraint that the same proportion of edges are deleted from each connected pair of vertex sets has become known as the Turán density problem. In Chapter 5, using entropy compression supplemented with some analytic methods, we derive an upper bound of 1 − 1/('y(Δ(H) − /3)), where Δ(H) is the maximum degree of H, 3 ≤ 'y < 4 and /3 ≤ 1. The new bound asymptotically approaches the existing best upper bound despite being derived in a completely different way. The techniques used in these results, illustrating their breadth and connec¬tions between them, are set out in Chapter 1