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. It was proved by Ben Green and independently by Alexander Sapozhenko in 2003.
See also Erdős conjecture
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