  There are many extremely challenging problems about existence of
monochromatic arithmetic progressions in colorings of groups. Many theorems
hold only for abelian groups as results on non-abelian groups are often much
more difficult to obtain. In this research project we do not only determine
existence, but study the more general problem of counting them. We formulate
the enumeration problem as a problem in real algebraic geometry and then use
state of the art computational methods in semidefinite programming and
representation theory to derive lower bounds for the number of monochromatic
arithmetic progressions in any finite group.
