Monomial Gotzmann sets

A homogeneous set of monomials in a quotient of the polynomial ring S := F[x1, . . . , xn] 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[x1, . . . , xn]/(x a 1 ) arise from certain Gotzmann sets in S. Then we partition the monomials in a Gotzmann set in S with respect to the multiplicity of xi 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[x1, . . . , xi−1, xi+1, . . . xn]. We also adopt some properties of the minimal growth of the Hilbert function in S to R.