English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The classical sharp threshold theorem of Friedgut and Kalai (1996) assertsthat any symmetric monotone function $f:\{0,1\}^{n}\to\{0,1\}$ exhibits a sharpthreshold phenomenon. This means that the expectation of $f$ with respect tothe biased measure $\mu_{p}$ increases rapidly from 0 to 1 as $p$ increases.  In this paper we present `robust' versions of the theorem, which assert thatit holds also if the function is `almost' monotone, and admits a much weakernotion of symmetry. Unlike the original proof of the theorem which relies onhypercontractivity, our proof relies on a `regularity' lemma (of the class ofSzemer\'edi's regularity lemma and its generalizations) and on the `invarianceprinciple' of Mossel, O'Donnell, and Oleszkiewicz which allows (under certainconditions) replacing functions on the cube $\{0,1\}^{n}$ with functions onGaussian random variables.  The hypergraph removal lemma of Gowers (2007) and independently of Nagle,R\"odl, Schacht, and Skokan (2006) says that if a $k$-uniform hypergraph on $n$vertices contains few copies of a fixed hypergraph $H$, then it can be made$H$-free by removing few of its edges. While this settles the `hypergraphremoval problem' in the case where $k$ and $H$ are fixed, the result ismeaningless when $k$ is large (e.g. $k>\log\log\log n$).  Using our robust version of the Friedgut-Kalai Theorem, we obtain ahypergraph removal lemma that holds for $k$ up to linear in $n$ for a largeclass of hypergraphs. These contain all the hypergraphs such that both theirnumber of edges and the sizes of the intersections of pairs of their edges areupper bounded by some constant.

Hypergraph Removal Lemmas via Robust Sharp Threshold Theorems