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 ${\displaystyle O{\big (}{2^{N/2}}{\big )}.}$

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