  One of the toughest problems in Ramsey theory is to determine the existence
of monochromatic arithmetic progressions in groups whose elements have been
colored. We study the harder problem to not only determine the existence of
monochromatic arithmetic progressions, but to also count them. We reformulate
the enumeration in real algebraic geometry and then use state of the art
computational methods in semidefinite programming and representation theory to
derive sharp, or an explicit constant from sharp, lower bounds for the cyclic
group of any order.
