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Updated Oct 7

Problem 1186

claims/: The 2 claim pages of Problem 1186, one per claimant's result; the problem's standing derives from them.

Statement. Let $\delta_k$ be such that in any $2$ -colouring of $\{1,\ldots,n\}$ there exist at least $(\delta_k+o(1))n^2$ many monochromatic $k$ -term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\delta_k$ .

Status. Open. The label is the site's (OPEN, page last edited 8 April 2026); its commentary records the bounds $1675/32768\le\delta_3\le117/2192$ of Parrilo, Robertson and Saracino, an accepted partial claim on its claim page. The site's proof-claims tab carries a partial proof claim by Carlos Toledo, submitted 2026-10-05 and produced with Claude (Anthropic), the system the claim names, that $\delta_3=117/2192$ , so that the upper bound of Parrilo, Robertson and Saracino is exact, by a computer-assisted proof closed with exact rational certificates; it is recorded as claimed on its claim page.

Source. erdosproblems.com/1186, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1186, https://www.erdosproblems.com/1186.

References.

[CCS07] Cameron, Peter and Cilleruelo, Javier and Serra, Oriol, On monochromatic solutions of equations in groups. Rev. Mat. Iberoam. (2007), 385-395.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[LuPe12] Lu, Linyuan and Peng, Xing, Monochromatic 4-term arithmetic progressions in 2-colorings of $\Bbb Z_n$ . J. Combin. Theory Ser. A (2012), 1048-1065.
[PRS08] Parrilo, Pablo A. and Robertson, Aaron and Saracino, Dan, On the asymptotic minimum number of monochromatic 3-term arithmetic progressions. J. Combin. Theory Ser. A (2008), 185-192.
[Wo10] Wolf, J., The minimum number of monochromatic 4-term progressions in $\Bbb Z_p$ . J. Comb. (2010), 53-68.

Formalization. None recorded.

Current assessment

Open; $\delta_3$ is bounded, and its claimed exact value is pending. For $k=3$ the refereed bounds $1675/32768\le\delta_3\le117/2192$ of Parrilo, Robertson and Saracino are an accepted partial claim on its claim page, and Toledo's computer-assisted claim that the upper bound is exact is pending on its claim page. The results of Cameron, Cilleruelo and Serra [CCS07], Wolf [Wo10] and Lu and Peng [LuPe12] that the commentary records get no claim page, since they bound the analogue $\tilde\delta_k$ for colorings of $\mathbb{Z}/p\mathbb{Z}$ , not $\delta_k$ . Lu and Peng also carry their construction over to $\{1,\ldots,n\}$ , giving a coloring with a third fewer monochromatic $4$ -term progressions than a random one, so $\delta_4\le1/72$ ; the site does not credit that bound, and it is recorded here without a claim page. No release item or lead names the problem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (23)

Parrilo, Robertson and Saracino's bounds for delta_3 Toledo's computer-assisted value of delta_3 problems/additive_combinatorics/E1186/claims Sumsets and Arithmetic Progressions additive_combinatorics/lu_2012_monochromatic_4_term_arithmetic_progressions_2 Conjecture 1 (p. 5): the infimum of m_4(Z_n) over n not divisible by 4 is 1/12 Conjecture 2 (p. 6): for k >= 4, limsup m_k([n]) equals liminf m_k(Z_n)Lemma 1 (p. 5): limsup m_k([n]) <= m_k(Z_b) for every b, giving c_4 <= 1/72 and c_5 <= 1/304 Theorem 1 (p. 4): for large primes p, 7/96 <= m_4(Z_p) <= 17/150 + o(1)Theorem 2 (p. 4): m_4(Z_n) >= 7/96 if 4 does not divide n and >= 2/33 if it does Theorem 3 (p. 4): m_4(Z_n) <= 17/150 + o(1) for odd n and <= 8543/72600 + o(1) for even n Theorem 4 (p. 5): m_5(Z_n) <= 3629/65712 + o(1) for odd n and <= 3647/65712 + o(1) for even n Theorem 5 (p. 5): liminf m_4(Z_n) <= 1/12 and liminf m_5(Z_n) <= 1/38 Theorem 6 (p. 6): every 2-coloring of Z_n has at least n^2/4 monochromatic 3-term progressions additive_combinatorics/parrilo_2008_asymptotic_minimum_number_monochromatic_3_term Lemma 1: a bound |N^+| <= cn^2(1+o(1)) gives V(S_0,S_1) >= (1/2)(3/8 - c)n^2(1+o(1))Theorem 2: V(n) >= (189/4096)n^2(1+o(1)) by enumeration with sixteen strips Theorem 4: every 2-coloring of [1,n] has at least (1675/32768)n^2(1+o(1)) monochromatic 3-term progressions Theorem 5: a twelve-block 2-coloring of [1,n] gives V(n) <= (117/2192)n^2(1+o(1))additive_combinatorics/wolf_2010_minimum_number_monochromatic_4_term_progressions Lemma 2.1 (p. 56): an identity between the counts of 4-term progressions by number of red elements Theorem 1.1 (p. 54): every 2-coloring of Z_p has at least p^2/32 monochromatic 4-term progressions Theorem 1.2 (p. 55): a 2-coloring of Z_p with fewer than (1 - 1/259200)p^2/16 monochromatic 4-term progressions

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