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In combinatorics, the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in $$|N|=\{1,\ldots,N\} is O\left({2^{N/2}}\right)$$.

The sum of two odd numbers is even, so a set of odd numbers is always sum-free. There are $$\lceil N/2\rceil$$odd numbers in |N|, and so $$2^{N/2}$$ subsets of odd numbers in |N|. The Cameron–Erdős conjecture says that this counts a constant proportion of the sum-free sets.

The conjecture was stated by Peter Cameron and Paul Erdős in 1988.[1] It was proved by Ben Green[2] and independently by Alexander Sapozhenko[3][4] in 2003.

Erdős conjecture

Notes

Cameron, P. J.; Erdős, P. (1990), "On the number of sets of integers with various properties", Number theory: proceedings of the First Conference of the Canadian Number Theory Association, held at the Banff Center, Banff, Alberta, April 17-27, 1988, Berlin: de Gruyter, pp. 61–79, MR 1106651.
Green, Ben (2004), "The Cameron-Erdős conjecture", The Bulletin of the London Mathematical Society 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752.
Sapozhenko, A. A. (2003), "The Cameron-Erdős conjecture", Doklady Akademii Nauk 393 (6): 749–752, MR 2088503.
Sapozhenko, Alexander A. (2008), "The Cameron-Erdős conjecture", Discrete Mathematics 308 (19): 4361–4369, doi:10.1016/j.disc.2007.08.103, MR 2433862.

Mathematics Encyclopedia