Cameron–Erdős conjecture

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

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