Monomial Gotzmann sets

A homogeneous set of monomials in a quotient of a 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 obtain bounds on the size of a component in the partition depending on neighboring components. We 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.