A Hilton–Milner-type theorem and an intersection conjecture for signed sets

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, . . . , n} and any integer k ≥ 2, let Sn,r,k be the family {{(x1, y1), . . . , (xr, yr)}: x1, . . . , xr are distinct elements of [n], y1, . . . , yr ∈ [k]} of k-signed r-sets on [n]. Let m := max{0, 2r−n}.We establish the following Hilton–Milner-type theorems, the second of which is proved using the first: (i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1| + |A2| ≤ n R K r −r i=m r I (k − 1) I n – r r – I K r−i + 1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2 ≤ r ≤ n, then |A| ≤ n – 1 r – 1 K r−1 −r−1 i=m r I (k − 1) I n − 1 – r r − 1 – I K r−1−i + 1 if r < n; k r−1 − (k − 1) r−1 + k − 1 if r = n. We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.peer-reviewe

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