On the number and structure of sum-free sets in a segment of positive integers

Omelyanov, K. G.; Sapozhenko, A. A.

doi:10.1515/156939203322733345

Cited by 5 publications

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“…Theorem 1 gives upper bound for the number of independent sets. Then using the results from [8,3], we prove that S(n) has almost correct system of containers.…”

Section: Introductionmentioning

confidence: 91%

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The Cameron–Erdős conjecture

Sapozhenko

2008

Discrete Mathematics

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“…Theorem 1 gives upper bound for the number of independent sets. Then using the results from [8,3], we prove that S(n) has almost correct system of containers.…”

Section: Introductionmentioning

confidence: 91%

“…The proof does not use the results from [3]. Later in [8] the estimate s(t, n) = O(2 n/2 ) was proved for t = n 3/4 log 2 n. Moreover, in [8] the structure of sum-free sets was described (see Theorem 3 below). Let S 1 (n) be the family of all subsets of odd numbers from [1, n] and s 1 (n) = |S 1 (n)| = 2 n/2 .…”

Section: Introductionmentioning

confidence: 99%

The Cameron–Erdős conjecture

Sapozhenko

2008

Discrete Mathematics

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“…Bounds for the number of independent sets in Cayley graphs were used in [2], [3], and [4] for the proof of the Cameron-Erdős conjecture about the number of sum-free sets in the segment OE1; n.…”

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confidence: 99%

On the number of independent sets in damaged Cayley graphs

Omelyanov¹

2005

Discrete Mathematics and Applications

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The Cayley graph generated by a set A is the graph A .V / on a set of positive integers V such that a pair .u; v/ 2 V V is an edge of the graph if and only if ju vj 2 A or uCv 2 A. We denote by G 2 .n; m/ the class of graphs G D .V; E/ such that G is a union of chains and cycles and jV j D n, jEj D m. In this paper, we present an upper bound for the number of independent sets in Cayley graphs A .V / such that A D fdn=2e t; dn=2e f g, V Â OEbn=2c C 1; bn=2c C t [ OEn t C 1; n, where n; t; f 2 N and f < t < n=4. We also describe the graph with the maximum number of independent sets in the family G 2 .n; m/.This research was supported by the Russian Foundation for Basic Researches, grant 04-01-00359.A subset C of vertices of a graph G is called independent if the subgraph generated by C has no edges. We denote by I.G/ the number of all independent sets in a graph G. For any real numbers p and q, we denote by OEp; q the set of natural numbers x such that p Ä x Ä q. The number of vertices in a chain (cycle) is called the length of this chain (cycle), an isolated vertex is considered as a chain of length one. We denote by G 2 .n; m/ the family of graphs G D .V; E/ such that G is a union of chains and cycles and jV j D n, jEj D m. We denote by A .V / the Cayley graph on the set of natural numbers V generated by a set A, that is, such that any pair .u; v/ 2 V V is an edge of the graph if and only if ju vj 2 A or u C v 2 A.Cayley graphs were used in [1] in order to determine the number of sum-free sets in groups. Bounds for the number of independent sets in Cayley graphs were used in [2], [3], and [4] for the proof of the Cameron-Erdős conjecture about the number of sum-free sets in the segment OE1; n.The Cayley graphs that are considered in this paper emerged during the computation of the constants in the Cameron-Erdős conjecture. Theorem 1 gives an upper bound for the number of independent sets in the Cayley graphs of some special form. This is the main result of the paper. We describe also the graph with the maximum number of independent sets in the family G 2 .n; m/. Theorem 2 partially (for k D 2) confirms the N. Alon conjecture about the structure of the k-regular graphs with the maximum number of independent sets. Theorem 1. Let n; t; f 2 N, f < t < n=4, and A D fdn=2e t; dn=2e f g;V D OEbn=2c C 1; bn=2c C t [ OEn t C 1; n:

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“…Under these mappings, sum-free sets are mapped into sum-free sets. Hence the required assertion follows if we take into account the equalities s.n=2 C t C 1; n 2t 1/ D 2s.n=2 C t; n 2t 3/ for even n and s.dn=2e C t C 1; n 2t 2/ D 2s.dn=2e C t; n 2t 4/ for odd n. It follows from Theorem 3 in [8] that O s.bn=6c; n/ D o.2 n=2 /. This relation and equalities (11), (12), (17), (18), and (19) imply that…”

Section: Theorem 2 the Cameron-erdős Constants Satisfy The Inequalitiesmentioning

confidence: 79%

“…In [7], it is proved that s.n=4; n/ D O.2 n=2 /. In [8], it is shown that s.q; n/ D O.2 n=2 / for q n 3=4 p log n. The Cameron-Erdős hypothesis was proved independently by B. Green [9] and A.…”

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confidence: 95%

Estimates of the Cameron–Erdős constants

Omelyanov¹

2006

Discrete Mathematics and Applications

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A set B of integers is called sum-free if for any a; b 2 B the number a C b does not belong to the set B. Let s.n/ be the number of sum-free sets in the interval of natural numbers OE1; n. As shown by Cameron, Erdős, and Sapozhenko, there exist constants c 0 and c 1 such that s.n/ .c 0 C 1/2 dn=2e for even n and s.n/ .c 1 C 1/2 dn=2e for odd n tending to infinity. The constants c 0 and c 1 are usually referred to as the Cameron-Erdős constants. In this paper, we obtain upper and lower bounds for the Cameron-Erdős constants which give the two first decimal places of their exact values.

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On the number and structure of sum-free sets in a segment of positive integers

Cited by 5 publications

References 4 publications

The Cameron–Erdős conjecture

The Cameron–Erdős conjecture

On the number of independent sets in damaged Cayley graphs

Estimates of the Cameron–Erdős constants

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