Gotzmann monomial ideals

A Gotzmann monomial ideal of the polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann’s persistence theorem. A subset $V$ 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 $V$ with $\mathrm{\mid }V\mathrm{\mid }=a$ is lexsegment (up to the permutation of the variables). In addition, we classify all Gotzmann subsets of $K[x_1,x_2,x_3]$.