Journal ArticleDOI
A Szemeredi-type regularity lemma in abelian groups, with applications
Reads0
Chats0
TLDR
In this article, an analogue of Szemeredi's regularity lemma in the context of abelian groups is presented and used to derive some results in additive number theory.Abstract:
Szemeredi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If
$$A \subseteq \{1,\ldots,N\}$$ has δ N2 triples (a1, a2, a3) for which a1 + a2 = a3 then A = B ∪ C, where B is sum-free and |C| = δ′N, and
$$\delta^{\prime} \rightarrow 0$$
as
$$\delta \rightarrow 0.$$
Another answers a question of Bergelson, Host and Kra. If
$$\alpha, \epsilon > 0,$$
if
$$N\,>\,N_{0}(\alpha, \epsilon)$$
and if
$$A \subseteq \{1,\ldots,N\}$$
has size α N, then there is some d ≠ 0 such that A contains at least
$$(\alpha^{3}-\epsilon)N$$
three-term arithmetic progressions with common difference d.read more
Citations
More filters
Journal ArticleDOI
The primes contain arbitrarily long arithmetic progressions
Benjamin W. Green,Terence Tao +1 more
TL;DR: In this paper, it was shown that there are arbitrarily long arithmetic progressions of primes and that a large fraction of the primes can be placed inside a pseudorandom set of almost primes with positive relative density.
Journal ArticleDOI
Hypergraph containers
David Saxton,Andrew Thomason +1 more
TL;DR: The notion of containment for independent sets in hypergraphs was introduced in this article, where it was shown that for a given hypergraph G, there exists a relatively small collection of vertex subsets, such that every independent set of G is contained within a member of the subset, and no member is large; the collection reveals an underlying structure to the independent sets.
Journal ArticleDOI
A new proof of the graph removal lemma
TL;DR: In this paper, the authors give a new proof which avoids Szemer edi's regularity lemma and gives a better bound for the directed and multicolored analogues of the graph removal lemma.
Journal ArticleDOI
Sum-free sets in abelian groups
Ben Green,Imre Z. Ruzsa +1 more
TL;DR: In this article, the maximal density of a sum-free subset of an abelian group was shown to be 2(μ(G)+o(1))n, which is tight up to theo-term.
Journal ArticleDOI
Regular Partitions of Hypergraphs: Regularity Lemmas
Vojtech Rödl,Mathias Schacht +1 more
TL;DR: The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.
References
More filters
Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Regular partitions of graphs
TL;DR: In this article, the authors generalize this result to arbitrary graphs, at the same time strengthening and simplifying the original bipartite result and showing that k-term arithmetic progression-free sets of integers must have density zero.
Szemeredi''s Regularity Lemma and its applications in graph theory
János Komlós,Miklós Simonovits +1 more
TL;DR: In this paper, the authors describe some typical applications and some generalizations of the regularity lemma, and also some new variants and generalizations appeared, and describe typical applications of the lemma.
Journal ArticleDOI
A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four
TL;DR: In this paper, the authors proposed a method to solve the problem of the problem: without abstracts, without abstractions, and without abstracting abstracts. But without abstract.