The minimum number of monochromatic 4-term progressions in \(\mathbb Z_p\)
The author improves the lower bound given by \textit{P. Cameron, J. Cilleruelo} and \textit{O. Serra} [Rev. Mat. Iberoam. 23, No. 1, 385--395 (2007; Zbl 1124.05086)] for the minimum number of monochromatic 4-term progressions contained in any 2-coloring of \(\mathbb Z_p\) with \(p\) a prime to \(\frac{p^2}{32}\). She also exhibits a coloring with significantly fewer than \(\frac{(1-\frac{1}{259200})p^2}{16}\) monochromatic 4-term progressions, which is based on a recent example in additive combinatories by Gowers. In the second half of this paper she discusses the corresponding problem in graphs, which has received a great deal more attention to date. She gives 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 \textit{G. Giraud} [J. Comb. Theory, Ser. B 27, 237--253 (1979; Zbl 0427.05052)], and briefly discusses the analogy between the upper-bound graph constructions of Thomason and hers for subsets of \(\mathbb Z_p\).
scientific article; zbMATH DE number 5776204