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Monomial Gotzmann sets in a quotient by a pure power

A homogeneous set of monomials in a quotient of the polynomial ring $S:=F[x_1, \..., x_n]$ is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient $R:=F[x_1, \..., x_n]/(x_1^a)$ arise from certain Gotzmann sets in $S$. Then we partition the monomials in a Gotzmann set in $S$ with respect to the multiplicity of $x_i$ and show that if the growth of the size of a component is larger than the size of a neighboring component, then this component is a multiple of a Gotzmann set in $F[x_1, \..., x_{i-1}, x_{i+1}, \...,x_n]$. We also adopt some properties of the minimal growth of the Hilbert function in $S$ to $R$.