Wolf 2010 minimum number monochromatic 4 term progressions

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lemma_2_1: States that in a 2-coloring of Z_p whose red class has size αp, the normalized counts c_i of 4-term progressions with exactly i red elements satisfy 4(c_0 + c_4) + (c_1 + c_3) = 4(1 - 3α + 3α^2).

theorem_1_1: States that for p prime every 2-coloring of Z_p contains at least p^2/32 monochromatic 4-term arithmetic progressions, that is m_4 >= 1/16 + o(1).

theorem_1_2: States that there is a 2-coloring of Z_p with fewer than (1 - 1/259200)p^2/16 monochromatic 4-term progressions, so that m_4 <= (1/8)(1 - 1/259200) + o(1), below the random count.

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Wolf, J., The minimum number of monochromatic 4-term progressions in {$\Bbb Z_p$}. J. Comb. 1 (2010), no. 1, 53--68.

Wolf studies M_4(p), the least number of monochromatic 4-term arithmetic progressions in a 2-coloring of Z_p, normalized as m_4 = 2M_4(p)/p^2. Theorem 1.1 proves that every 2-coloring contains at least p^2/32 monochromatic 4-APs, i.e. m_4 >= 1/16 + o(1), improving the bounds 1/185 from van der Waerden averaging and 1/20 and 2/33 of Cameron, Cilleruelo and Serra. Their identity relating the numbers of 4-APs with each number of red elements (Lemma 2.1, borrowed from them with a new double-counting proof) leaves a term depending on the red density a, which they discard; the proof keeps it and adds a lower bound a(1 - a) on the proportion of evenly colored 4-APs, found by comparing the two 4-APs that extend each 3-term progression, so that m_4 >= (a(1 - a) + 3(1 - 2a)^2)/4 >= 1/16 (Section 2). In the other direction a random coloring gives m_4 <= 1/8 + o(1), and Theorem 1.2, proved in Section 3, gives a 2-coloring with fewer than (1 - 1/259200)p^2/16 monochromatic 4-APs, i.e. m_4 <= (1/8)(1 - 1/259200) + o(1). The coloring comes from a recent example of Gowers of a uniform set with fewer 4-APs than a random set of the same density; its last step rounds a bounded function to a set at random, so the coloring is shown to exist rather than written down. The second half of the paper treats the graph analog, giving a simplified proof of Giraud's best known lower bound on monochromatic K_4's in 2-colorings of K_n and comparing Thomason's upper-bound graph constructions with the Z_p constructions here. Problem 1186 asks how many monochromatic k-term progressions every 2-coloring of {1,...,n} must contain; the paper treats only the cyclic-group analogue for k = 4, where it shows that random colorings of Z_p do not minimize the count, and it names the interval {1,...,n} as a separate question (p. 55).

Source: https://doi.org/10.4310/joc.2010.v1.n1.a4. No notice is printed on pp. 53--54 or 67--68, the Crossref record names no license, and the publisher's page could not be read on 2026-10-02 (the DOI resolves to link.intlpress.com, which answered HTTP 403, as did the International Press journal site); the term is unstated.

Results.

• Theorem 1.1 (p. 54): for $p$ prime, every 2-coloring of $\mathbb Z_p$ contains at least $p^2/32$ monochromatic 4-term progressions, that is $m_4\ge1/16+o(1)$. The intermediate display on p. 56 has $\tfrac12E$ where the identity before it gives $\tfrac14E$; the result page explains why this is read as a misprint that leaves the bound unaffected.
• Theorem 1.2 (p. 55): there is a 2-coloring of $\mathbb Z_p$ with fewer than $(1-1/259200)p^2/16$ monochromatic 4-term progressions, against the random count $p^2/16$, that is $m_4\le\frac18(1-\frac1{259200})+o(1)$; Section 3 builds it from an example of Gowers, with a random rounding step.
• Lemma 2.1 (p. 56): for a 2-coloring of $\mathbb Z_p$ whose red class has size $\alpha p$, the numbers $c_i$ of 4-term progressions with exactly $i$ red elements, divided by $p^2/2$, satisfy $4(c_0+c_4)+(c_1+c_3)=4(1-3\alpha+3\alpha^2)$; the paper borrows it from Cameron, Cilleruelo and Serra and gives its own double-counting proof.

Read status: claims checked. The statements of Theorems 1.1 and 1.2 and Lemma 2.1 were read clause by clause against the print, the proofs of Section 2 were followed through, and the construction of Section 3 was read for its structure only; no proof was independently verified.

Bears on. #1186: background only. The problem asks for the least normalized number $\delta_k$ of monochromatic $k$-term progressions in 2-colorings of $\{1,\ldots,n\}$; Theorems 1.1 and 1.2 bound the analogous quantity $m_4$ for the cyclic group $\mathbb Z_p$, $p$ prime, and the paper derives no bound on $\delta_4$. It names the interval question as a separate one (p. 55).

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