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,...,xn] denote the polynomial ring in n variables over a field K and Md the set of monomials of R of degree d. A subset V ⊂ Md 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 ⊂ Md with |V| = a is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of K[x1, x2, x3].
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