41 research outputs found

    Global hypercontractivity and its applications

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    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 pp-biased measures. However, simple examples show that when p=o(1)p = o(1), there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general pp 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 pp-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

    Covering and tiling hypergraphs with tight cycles

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    Given 3≤k≤s3 \leq k \leq s, we say that a kk-uniform hypergraph CskC^k_s is a tight cycle on ss vertices if there is a cyclic ordering of the vertices of CskC^k_s such that every kk consecutive vertices under this ordering form an edge. We prove that if k≥3k \ge 3 and s≥2k2s \ge 2k^2, then every kk-uniform hypergraph on nn vertices with minimum codegree at least (1/2+o(1))n(1/2 + o(1))n has the property that every vertex is covered by a copy of CskC^k_s. Our result is asymptotically best possible for infinitely many pairs of ss and kk, e.g. when ss and kk are coprime. A perfect CskC^k_s-tiling is a spanning collection of vertex-disjoint copies of CskC^k_s. When ss is divisible by kk, the problem of determining the minimum codegree that guarantees a perfect CskC^k_s-tiling was solved by a result of Mycroft. We prove that if k≥3k \ge 3 and s≥5k2s \ge 5k^2 is not divisible by kk and ss divides nn, then every kk-uniform hypergraph on nn vertices with minimum codegree at least (1/2+1/(2s)+o(1))n(1/2 + 1/(2s) + o(1))n has a perfect CskC^k_s-tiling. Again our result is asymptotically best possible for infinitely many pairs of ss and kk, e.g. when ss and kk are coprime with kk even.Comment: Revised version, accepted for publication in Combin. Probab. Compu

    Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

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    A graph is OkO_k-free if it does not contain kk pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in OkO_k-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) OkO_k-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of O2O_2-free graphs without K3,3K_{3,3}-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse OkO_k-free graphs, and that deciding the OkO_k-freeness of sparse graphs is polynomial time solvable.Comment: 28 pages, 6 figures. v3: improved complexity result

    Subject Index Volumes 1–200

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    Extremal Combinatorics

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