The minimum number of monochromatic 4-term progressions in Z(p)

In this paper we improve the lower bound given by Cameron, Cilleruelo and Serra for the minimum number of monochromatic 4-term progressions contained in any 2-coloring of Z(p) with p a prime. We also exhibit a coloring with significantly fewer than the random number of monochromatic 4-term progressions, which is based on an a recent example in additive combinatorics by Gowers. In the second half of this paper we discuss the corresponding problem in graphs, which has received a great deal more attention to date. We give a simplified proof of the best known lower bound on the minimum number of monochromatic K(4)s contained in any 2-coloring of K-n by Giraud, and briefly discuss the analogy between the upper-bound graph constructions of Thomason and ours for subsets of Z(p).