In combinatorics , the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in
[
N
]
=
{
1
,
… … -->
,
N
}
{\displaystyle [N]=\{1,\ldots ,N\}}
is
O
(
2
N
/
2
)
.
{\displaystyle O{\big (}{2^{N/2}}{\big )}.}
The sum of two odd numbers is even , so a set of odd numbers is always sum-free. There are
⌈ ⌈ -->
N
/
2
⌉ ⌉ -->
{\displaystyle \lceil N/2\rceil }
odd numbers in [N ], and so
2
N
/
2
{\displaystyle 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
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, ISBN 9783110117233 , 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 , S2CID 119615076 .
^ 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 .