26 research outputs found
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
On the structure of graphs with forbidden induced substructures
One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.
Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every -edge-colouring of the complete graph on vertices there is a monochromatic clique on at least vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.
In the second part of this thesis we focus more on order-size pairs; an order-size pair is the family consisting of all graphs of order and size , i.e. on vertices with edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs , i.e. for approaching infinity, the limit superior of the fraction of all possible sizes , such that the order-size pair does not avoid the pair