37 research outputs found
High dimensional Hoffman bound and applications in extremal combinatorics
One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on -wise intersecting families
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 -biased measures. However, simple examples show
that when , there is no hypercontractive inequality that is strong
enough.
In this paper, we establish an effective hypercontractive inequality for
general 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
-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
Discrete Geometry and Convexity in Honour of Imre Bárány
This special volume is contributed by the speakers of the Discrete Geometry and
Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference
is to celebrate the 70th birthday and the scientific achievements of professor
Imre Bárány, a pioneering researcher of discrete and convex geometry, topological
methods, and combinatorics. The extended abstracts presented here are written by
prominent mathematicians whose work has special connections to that of professor
Bárány. Topics that are covered include: discrete and combinatorial geometry,
convex geometry and general convexity, topological and combinatorial methods.
The research papers are presented here in two sections. After this preface and a
short overview of Imre Bárány’s works, the main part consists of 20 short but very
high level surveys and/or original results (at least an extended abstract of them)
by the invited speakers. Then in the second part there are 13 short summaries of
further contributed talks.
We would like to dedicate this volume to Imre, our great teacher, inspiring
colleague, and warm-hearted friend