The minimum number of monochromatic 4-term progressions

It is not difficult to see that whenever you 2-color the
elements of Z/pZ, the number of monochromatic 3-term arithmetic
progressions depends only on the density of the color classes. The
analogous statement for 4-term progressions is false. We shall analyse
the reasons for this, and subsequently derive bounds on the minimum
number of monochromatic 4-term arithmetic progressions in any 2-coloring
of Z/pZ. In the process we touch upon the subject of quadratic Fourier
analysis as well as a closely related question in graph theory studied
by Thomason et al.: What is the minimum number of monochromatic
K_4s in any 2-coloring of K_n?