Gotzmann Monomials In Four Variables

It is a widely open problem to determine which monomials in the n-variable polynomial ring $K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ over a field $K$ have the Gotzmann property, 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.