On minimal presentations of shifted affine semigroups with few generators

An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, and numerical an with $d = 1$. A growing body recent work examines shifted families semigroups, that is, semigroups the form $M_n \langle n + r_1, \ldots, r_k \rangle$ for fixed $r_1, r_k$, one each value shift parameter $n$. It has been shown within any family size minimal presentation bounded (in fact, this eventually periodic in $n$). In paper, we consider demonstrate some, but not all, 4-generated have arbitrarily large presentations.