On tt-Intersecting Families of Permutations

We prove that there exists a constant $c_0$ such that for any $t \in \mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge 0.75(n-t)!$ then there exist $i_1,\ldots,i_t$ and $j_1,\ldots,j_t$ such that $\sigma(i_1)=j_1,\ldots,\sigma(i_t)=j_t$ holds for any $\sigma \in A$. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on $t$-intersecting families of permutations hold for all $t \leq c_0 n$. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to $t$-intersecting sub-families of `pseudorandom' families in $\{1,2,\ldots,n\}^n$, like $S_n$

Extremal sets with forbidden configurations and the independence ratio of geometric hypergraphs

No abstrac

On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Given a set of n points in R^d, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the l_p-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d=omega(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS\u2717], Williams [SODA\u2718], David-Karthik-Laekhanukit [SoCG\u2718]). In this paper, we show that for every p in R_{>= 1} cup {0}, under the Strong Exponential Time Hypothesis (SETH), for every epsilon>0, the following holds: - No algorithm running in time O(n^{2-epsilon}) can solve the Closest Pair problem in d=(log n)^{Omega_{epsilon}(1)} dimensions in the l_p-metric. - There exists delta = delta(epsilon)>0 and c = c(epsilon)>= 1 such that no algorithm running in time O(n^{1.5-epsilon}) can approximate Closest Pair problem to a factor of (1+delta) in d >= c log n dimensions in the l_p-metric. In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n^{epsilon} factor in the running time) for d=(log n)^{Omega_epsilon(1)} dimensions. Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n^{o(1)}-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product. At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n^{2-epsilon} edges whose vertices can be realized as points in a (log n)^{Omega_epsilon(1)}-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory\u2703]

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