Sharp bound on the number of maximal sum-free subsets of integers

Balogh, József; Liu, Hong; Sharifzadeh, Maryam; Treglown, Andrew

doi:10.4171/jems/802

Cited by 34 publications

(74 citation statements)

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“…A similar result for sum-free sets was obtained by Balogh, Liu, Sharifzadeh, and Treglown [11,12], who determined the number of maximal sum-free subsets of {1, . .…”

Section: List Colouringsupporting

confidence: 71%

“…Additive combinatorics. The method of hypergraph containers has been applied to a number of different number-theoretic objects, including sum-free sets [6,7,11,12], Sidon sets [82], sets containing no k-term arithmetic progression [10,13], and general systems of linear equations [82]. (See also [49,50,78] for early applications of the container method to sum-free sets and [28,29,27,59] for applications of graph containers to B h -sets.)…”

Section: List Colouringmentioning

confidence: 99%

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The Method of Hypergraph Containers

Balogh

Morris

Samotij

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.

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“…A similar result for sum-free sets was obtained by Balogh, Liu, Sharifzadeh, and Treglown [11,12], who determined the number of maximal sum-free subsets of {1, . .…”

Section: List Colouringsupporting

confidence: 71%

Section: List Colouringmentioning

confidence: 99%

The Method of Hypergraph Containers

Balogh

Morris

Samotij

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…For future reference, we note the following two simple corollaries of Observation A.2 and our assumptions on the maximum degrees of H, see (2). Suppose that (i 0 , i 1 ) ∈ U.…”

Section: Typical Structure Resultsmentioning

confidence: 99%

On the number of sets with a given doubling constant

Campos

2020

Isr. J. Math.

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We study the number of s-element subsets J of a given abelian group G, such that |J + J| ≤ K|J|. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for K fixed, we provide an upper bound on the number of such sets which is tight up to a factor of 2 o(s) , when G = Z and K = o(s/(log n) 3 ). We also provide a generalization of this result to arbitrary abelian groups which is tight up to a factor of 2 o(s) in many cases. The main tool used in the proof is the asymmetric container lemma, introduced recently by Morris, Samotij and Saxton. 2 δs 1 2 Ks s sets J ⊂ [n] with |J| = s and |J + J| ≤ K|J|.The conjecture was later confirmed for K constant by Green and Morris [7]; in fact they proved a slightly more general result: for each fixed K and as s → ∞, the number Research partially supported by CNPq. 1 of sets J ⊂ [n] with |J| = s and |J + J| ≤ K|J| is at most 2 o(s) 1 2 Ks s n ⌊K+o(1)⌋ .The authors of [7] used this result to bound the size of the largest clique in a random Cayley graph and recently the result was also applied by Balogh, Liu, Sharifzadeh and Treglown [2] to determine the number of maximal sum-free sets in [n]. Our main theorem confirms Conjecture 1.1 for all K = o(s/(log n) 3 ). Theorem 1.2. Let s, n be integers and 2 ≤ K ≤ o s (log n) 3 . The number of sets J ⊂ [n] with |J| = s such that |J + J| ≤ K|J| is at most 2 o(s) 1 2 Ks s .We will in fact prove stronger bounds on the error term than those stated above, see Theorem 4.1. Nevertheless, we are unable to prove the conjecture in the range K = Ω(s/(log n) 3 ), and actually the conjecture is false for a certain range of values of s and K ≫ s/ log n. More precisely, for any integers n, s, and any positive numbers K, ǫ with min{s, n 1/2−ǫ } ≥ K ≥ 4 log(24C)s ǫ log n , there are at least n 2 K 4 Ks 8 s − K 4 ≥ CKs s sets J ⊂ [n] with |J| = s and |J + J| ≤ Ks. The construction 1 is very simple: let P be an arithmetic progression of size Ks/8 and set J = J 0 ∪ J 1 , where J 0 is any subset of P of size s − K/4, and J 1 is any subset of [n] \ P of size K/4. For convenience we provide the details in the appendix. Our methods also allow us to characterize the typical structure of an s-set with doubling constant K, and obtain the following result. Theorem 1.3. Let s, n be integers and 2 ≤ K ≤ o s (log n) 3 . For almost all sets J ⊂ [n]with |J| = s such that |J + J| ≤ K|J|, there is a set T ⊂ J such that J \ T is contained in an arithmetic progression of size 1+o(1) 2 Ks and |T | = o(s). In the case s = Ω(n) (and hence K = O(1)), this result was proved by Mazur [10]. We will provide better bounds for the error terms in Theorem 5.1, below.

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“…, and, as Ω(·) is completely additive, we have 2 | Ω( i∈ [3] a i b i ), contradicting (4.4). Thus G par is product-free.…”

Section: 32mentioning

confidence: 89%

The number of multiplicative Sidon sets of integers

Liu

Pach

2019

Journal of Combinatorial Theory, Series A

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A set S of natural numbers is multiplicative Sidon if the products of all pairs in S are distinct. Erdős in 1938 studied the maximum size of a multiplicative Sidon subset of {1, . . . , n}, which was later determined up to the lower order term: π(n) + Θ( n 3/4 (log n) 3/2 ). We show that the number of multiplicative Sidon subsets of(log n) 3/2 ) for a certain function T (n) ≈ 2 1.815π(n) which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determined. It resolves the enumeration problem for multiplicative Sidon sets initiated by Cameron and Erdős in the 80s.We also investigate its extension for generalised multiplicative Sidon sets. Denote by S k , k ≥ 2, the number of multiplicative k-Sidon subsets of {1, . . . , n}. We show that S k (n) = (β k + o(1)) π(n) for some β k we define explicitly. Our proof is elementary.1

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Sharp bound on the number of maximal sum-free subsets of integers

Cited by 34 publications

References 20 publications

The Method of Hypergraph Containers

The Method of Hypergraph Containers

On the number of sets with a given doubling constant

The number of multiplicative Sidon sets of integers

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