Cameron–Erdős conjecture

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

The sum of two odd numbers is even, so a set of odd numbers is always sum-free. There are ${\displaystyle \lceil N/2\rceil }$ 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.

See also

• Erdős conjecture

Notes

1. 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.
2. 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.
3. Sapozhenko, A. A. (2003), "The Cameron-Erdős conjecture", Doklady Akademii Nauk, 393 (6): 749–752, MR 2088503.
4. Sapozhenko, Alexander A. (2008), "The Cameron-Erdős conjecture", Discrete Mathematics, 308 (19): 4361–4369, doi:10.1016/j.disc.2007.08.103, MR 2433862.