Divsets, numerical semigroups and Wilf’s conjecture

Let 𝑆⊆ℕ be a numerical semigroup with multiplicity 𝑚=min⁡(𝑆∖{0}) and conductor 𝑐=max⁡(ℤ∖𝑆)+1. Let P be the set of primitive elements, i.e. minimal generators, of S, and let L be the set of elements of S which are smaller than c. Wilf's conjecture (1978) states that the inequality |𝑃|⁢|𝐿|≥𝑐 must hold. The conjecture has been shown to hold in case |𝑃|≥𝑚/2 by Sammartano in 2012, and subsequently in case |𝑃|≥𝑚/3 by the author in 2020. The main result in this paper is that Wilf's conjecture holds in case |𝑃|≥𝑚/4 when m divides c. The proof uses divsets X, i.e. finite divisor-closed sets of monomials, as abstract models of numerical semigroups, and proceeds with estimates of the vertex-maximal matching number of the associated graph 𝐺⁡(𝑋).