Journal of Combinatorics

Volume 1 (2010)

Number 1

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

Pages: 53 – 68

DOI: http://dx.doi.org/10.4310/JOC.2010.v1.n1.a4

Author

J. Wolf (Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, N.J., U.S.A.)

Abstract

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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