The minimum number of monochromatic 4-term progressions in $\mbb{Z}_p$

In this paper we improve the lower bound given by Cameron, Cilleruelo and Serra for theminimum 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 therandom number of monochromatic 4-term progressions, which is based on an a recent examplein additive combinatorics by Gowers. In the second half of this paper we discuss thecorresponding 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 ofmonochromatic $K_4$s contained in any $2$-coloring of $K_n$ by Giraud, and brieflydiscuss the analogy between the upper-bound graph constructions of Thomason and ours forsubsets of $\Z_p$.

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Published 1 January 2010