Bounds for syzygies of monomial curves

Let Γ ⊆ N be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of Γ which depends only on the width of Γ, that is, the difference between the largest and the smallest generator of Γ. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8-28]. Moreover, for 4- generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.