Gotzmann monomial ideals

A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let R = K[x₁,...,x_n] denote the polynomial ring in n variables over a field K and M^d the set of monomials of R of degree d. A subset V ⊂ M^d is said to be a Gotzmann subset if the ideal generated by V is a Gotzmann monomial ideal. In the present paper, we find all integers a > 0 such that every Gotzmann subset V ⊂ M^d with |V| = a is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of K[x₁, x₂, x₃].