Cameron–Erdős conjecture

The conjecture that the number of sum-free subsets of {1, …, n} grows on the order of 2^(n/2). Green and Sapozhenko proved it.

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Sum-free set: A set of integers containing no three elements x, y, z with x + y = z. The conjecture counts these sets inside the first n positive integers.
Peter Cameron: A British mathematician whose work includes permutation groups and combinatorics. Cameron formulated the counting problem jointly with Erdős.
Ben Green: A British mathematician known for work in additive combinatorics and analytic number theory. Green proved the conjecture with Sapozhenko.
Schur triple: A triple of positive integers x, y, z satisfying x + y = z. Sum-free sets exclude precisely these additive configurations.
Additive container theorem: A theorem that covers independent sets of a hypergraph with a relatively small family of structured supersets. Green and Sapozhenko’s proof bounds the count by organizing sum-free sets into containers.
Paul Erdős: A Hungarian mathematician known for prolific work in number theory and combinatorics. Erdős co-formulated the conjecture and helped establish its additive-combinatorial context.
Alexander Sapozhenko: A mathematician known for work on combinatorics and the enumeration of independent sets. Sapozhenko’s methods supplied the other part of the proof.
Behrend set: A large subset of an integer interval containing no nontrivial three-term arithmetic progression. It avoids arithmetic progressions rather than the equation x + y = z.
Hypergraph independent set: A vertex set in a hypergraph that contains no complete hyperedge. A sum-free set is independent in the hypergraph whose edges encode additive triples.
Enumerative combinatorics: The branch of mathematics concerned with counting discrete structures. The conjecture asks for the asymptotic number of sets satisfying an additive restriction.