Wilf’s conjecture for all numerical semigroups

Prove or disprove Wilf’s conjecture by establishing whether every numerical semigroup S satisfies the inequality (F(S) + 1 − g(S)) / (F(S) + 1) ≥ 1 / e(S), where F(S) is the Frobenius number, g(S) is the genus, and e(S) is the embedding dimension.

Background

The paper reviews core invariants of numerical semigroups—embedding dimension, Frobenius number, and genus—and recalls Wilf’s conjecture, which ties these three parameters via a density inequality. Despite substantial progress on related bounds and random models, Wilf’s conjecture remains a central open problem in the field.

The authors’ work focuses on probabilistic bounds in Erdős–Rényi-type random numerical semigroup models; however, Wilf’s conjecture concerns all numerical semigroups and stands independently of the random model considered here.