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
$R=K[x_1,\dots,x_n]$ 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 $V\subset M^d$ 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 $V\subset M^d$ with
$|V|=a$ is lexsegment (up to the permutations of the
variables). In addition, we classify all Gotzmann subsets of
$K[x_1,x_2,x_3]$.