Gotzmann monomials in four variables

It is a widely open problem to determine which monomials in the $n$-variable polynomial ring $K[x_1,dots,x_n]$ over a field $K$ have the Gotzmann property, emph{i.e.} induce a Borel-stable Gotzmann monomial ideal. Since 2007, only the case $n le 3$ was known. Here we solve the problem for the case $n=4$. The solution involves a surprisingly intricate characterization.