Counting sum-free sets in abelian groups

Alon, Noga; Balogh, József; Morris, Robert; Samotij, Wojciech

doi:10.1007/s11856-013-0067-y

Cited by 31 publications

(79 citation statements)

References 37 publications

(70 reference statements)

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“…A simple version of it, applicable to 3-uniform hypergraphs with ∆ 2 (H n ) = O(1), was proved in [1] and used in [1,2] to count sum-free subsets in Abelian groups and in the set [n].…”

Section: Stability Results For Sparse Setsmentioning

confidence: 99%

“…The systematic study of the Turán problem in G(n, p) was initiated by Haxell, Kohayakawa, and Luczak [33,34] and by Kohayakawa,Luczak,and Rödl [41], who posed the following problem. For a fixed graph H, determine necessary and sufficient conditions on a sequence p ∈ [0,1] N of probabilities such that, a.a.s., ex G(n, p n ),…”

Section: Turán's Problem In Random Graphs a Famous Theorem Of Erdős mentioning

confidence: 99%

“…. , u j−1 } ; ties are broken by giving preference to vertices which come earlier in the order chosen in (1). (3) The max-degree order on V (G) is (u 1 , .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Independent sets in hypergraphs

Balogh

Morris

Samotij

2014

J. Amer. Math. Soc.

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304

676

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Abstract. Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called 'sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses.In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural 'counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the wellknown conjecture of Kohayakawa, Luczak, and Rödl, a probabilistic embedding lemma for sparse graphs, for all 2-balanced graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, such as sparse random versions of Szemerédi's theorem, the Erdős-Stone Theorem and the Erdős-Simonovits Stability Theorem, and obtain their natural 'counting' versions, which in some cases are considerably stronger. We also obtain new results, such as a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and, as a consequence of the K LR conjecture, we extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. Similar results have been discovered independently by Saxton and Thomason.

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“…A simple version of it, applicable to 3-uniform hypergraphs with ∆ 2 (H n ) = O(1), was proved in [1] and used in [1,2] to count sum-free subsets in Abelian groups and in the set [n].…”

Section: Stability Results For Sparse Setsmentioning

confidence: 99%

Section: Turán's Problem In Random Graphs a Famous Theorem Of Erdős mentioning

confidence: 99%

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Independent sets in hypergraphs

Balogh

Morris

Samotij

2014

J. Amer. Math. Soc.

Self Cite

304

676

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“…As corollaries of this general theorem, we then derive several new results. In particular, we remove the assumption that H is (strictly) 2-balanced from the statement of Theorem 1.4, where we also improve the implicit probability estimate from n −ω (1) to exp(−Ω(n 2 p n )), which is asymptotically best possible. Finally, we remark that our approach removes the somewhat artificial condition p n ≪ 1 present in the general transference theorems of both Conlon and Gowers [4], and Schacht [21] (the case p n = Ω(1) in Theorems 1.3 and 1.4 did not follow directly from the respective transference theorem and required additional arguments).…”

Section: Introductionmentioning

confidence: 99%

Stability results for random discrete structures

Samotij

2012

Random Struct Algorithms

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Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. the theorem of Schacht can be applied in a more general setting and yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdős and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting that is somewhat more general and stronger than the one obtained by Conlon and Gowers. We then use this theorem to derive several new results, among them a random version of the Erdős-Simonovits stability theorem for arbitrary graphs. the main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.

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“…Also, in [15,230,231,268,319] informationtheoretic techniques are used to study sumset inequalities. Very recently, Alon et al [5,6], using graph-theoretic methods, studied sum-free sets of order m in finite Abelian groups, and also, sum-free subsets of the set [1, n]. Additive combinatorics problems in matrix rings is another active area of research [53,55,67,82,83,114,125,133,134,183,184,206,298].…”

Section: Introductionmentioning

confidence: 99%

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

Bibak

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define -perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

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Counting sum-free sets in abelian groups

Cited by 31 publications

References 37 publications

Independent sets in hypergraphs

Independent sets in hypergraphs

Stability results for random discrete structures

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

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