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| In [[combinatorics]], the '''Cameron–Erdős conjecture''' (now a theorem) is the statement that the number of [[sum-free set]]s contained in <math>|N|=\{1,\ldots,N\}</math> is <math>O\left({2^{N/2}}\right).</math>
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| The sum of two odd numbers is even, so a set of odd numbers is always sum-free. There are <math>\lceil N/2\rceil</math> odd numbers in |''N''|, and so <math>2^{N/2}</math> subsets of odd numbers in |''N''|. The Cameron–Erdős conjecture says that this counts a constant proportion of the sum-free sets.
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| The conjecture was stated by [[Peter Cameron (mathematician)|Peter Cameron]] and [[Paul Erdős]] in 1988.<ref>{{citation
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| | last1 = Cameron | first1 = P. J. | author1-link = Peter Cameron (mathematician)
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| | last2 = Erdős | first2 = P. | author2-link = Paul Erdős
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| | contribution = On the number of sets of integers with various properties
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| | location = Berlin
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| | mr = 1106651
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| | pages = 61–79
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| | publisher = de Gruyter
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| | title = Number theory: proceedings of the First Conference of the Canadian Number Theory Association, held at the Banff Center, Banff, Alberta, April 17-27, 1988
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| | url = http://books.google.com/books?id=68g0Ds4FNM0C&pg=PA61&lpg=PA61
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| | year = 1990}}.</ref> It was proved by [[Ben J. Green|Ben Green]]<ref>{{citation
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| | last = Green | first = Ben | author-link = Ben J. Green
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| | arxiv = math.NT/0304058
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| | doi = 10.1112/S0024609304003650
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| | issue = 6
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| | journal = The Bulletin of the London Mathematical Society
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| | mr = 2083752
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| | pages = 769–778
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| | title = The Cameron-Erdős conjecture
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| | volume = 36
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| | year = 2004}}.</ref> and independently by Alexander Sapozhenko<ref>{{citation
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| | last = Sapozhenko | first = A. A.
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| | issue = 6
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| | journal = Doklady Akademii Nauk
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| | mr = 2088503
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| | pages = 749–752
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| | title = The Cameron-Erdős conjecture
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| | volume = 393
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| | year = 2003}}.</ref><ref>{{citation
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| | last = Sapozhenko | first = Alexander A.
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| | doi = 10.1016/j.disc.2007.08.103
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| | issue = 19
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| | journal = Discrete Mathematics
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| | mr = 2433862
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| | pages = 4361–4369
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| | title = The Cameron-Erdős conjecture
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| | volume = 308
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| | year = 2008}}.</ref> in 2003.
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| ==See also==
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| * [[Erdős conjecture]]
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| ==Notes==
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| {{reflist}}
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| {{DEFAULTSORT:Cameron-Erdos conjecture}}
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| [[Category:Additive number theory]]
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| [[Category:Combinatorics]]
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| [[Category:Theorems in discrete mathematics]]
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| [[Category:Paul Erdős]]
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| {{combin-stub}}
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