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Claim. The case $k=3$ of Problem 1186 has the exact value

$$\delta_3=\lim_{n\to\infty}\frac{V(n)}{n^2}=\frac{117}{2192}=0.05337\ldots,$$

where $V(n)$ is the least number of monochromatic three-term arithmetic progressions over all two-colorings of $\{1,\ldots,n\}$: the upper bound of Parrilo, Robertson and Saracino, attained by their coloring in twelve blocks of relative lengths $28,6,28,37,59,116,116,59,37,28,6,28$ out of $548$ alternating in color, is the truth, as they conjectured; their lower bound was $1675/32768$. The write-up is C. Toledo, The twelve-block colouring is optimal: $\delta_3=117/2192$, Theorem 1 (draft v0.2, October 2026). It re-derives the counting lemma of Parrilo, Robertson and Saracino (source card), which gives $V(n)/n^2\ge1/16+Q^*/4-O(1/n)$ with

$$Q^*=\inf\Bigl\{\iint_{a/2\le b\le(1+a)/2}\varphi(a)\varphi(b)\,da\,db: \varphi\colon[0,1]\to[-1,1]\text{ measurable}\Bigr\},$$

so that the theorem follows from $Q^*\ge-5/137$, the value the twelve-block step function $\varphi^*$ attains; the paper claims more, that $\varphi^*$ and $-\varphi^*$ are the only minimizers. The argument expands the quadratic form around $\varphi^*$, where the first-order term is an explicit nonnegative piecewise-linear function vanishing exactly at the block edges, passes to a finite discretization whose first-order term matches that function exactly, splits the admissible functions into five regions according to their $L^1$ distance from $\varphi^*$, and settles each region with an exact rational certificate: the region containing $\varphi^*$ is closed with no slack, and every other region with slack at least $2.747\times10^{-4}$. The paper says the certificates were generated by the claimant's programs, checked by a verifier written separately from them by the same AI agent in the same session, and re-decided by a second verifier written in another language from the mathematics and the file format alone, sharing no code with either; both verifiers accept the five certificates and reject every planted forgery. The certificates and verifiers are archived at the Zenodo record and described on the report page linked above.

Submission note. Posted to erdosproblems.com as a proof claim by Carlos Toledo (account carlos_toledo) on 5 October 2026, giving "Claude (Anthropic)" as the AI used:

The case $k=3$: $\delta_3=117/2192$, so the twelve-block colouring of Parrilo, Robertson and Saracino (2008) is optimal, as they conjectured. By their counting lemma it suffices that $\iint_{a/2\le b\le(1+a)/2}\varphi(a)\varphi(b)\ge-5/137$ for every measurable $\varphi:[0,1]\to[-1,1]$. Expanding around the twelve blocks, the first-order term is an explicit $h\ge0$ vanishing exactly at the block edges; a discretisation built on $h$ loses nothing to first order, and five regions are closed by exact rational certificates. The case $k\ge4$ is not addressed. Notes: Produced with substantial AI assistance (Claude); I have read and checked the mathematics myself, line by line. The certificates are verified in exact rational arithmetic by two independent programs that share no code, and planted forgeries are rejected; they re-run in seconds. Not peer reviewed, not formalised in Lean. Certificates, verifiers and records: https://carlostoledo.co/reports/delta3.html (archived at doi:10.5281/zenodo.23171167).

Covers. The case $k=3$ alone: the exact value of $\delta_3$, which is the question Graham offered a prize for, and the conjecture of Parrilo, Robertson and Saracino. The paper claims nothing about $k\ge4$, about the asymptotic formula or bounds for general $k$ that the problem asks for, or about the analogue in $\mathbb{F}_p$ (the $\tilde\delta_k$ of the site's commentary, page last edited 8 April 2026), and says its method does not carry over to $k=4$, where the counting functional has a quartic term that no reduction of its kind removes.

Depends on. Nothing in this wiki; the reduction is re-derived in the paper.

Standing. Claimed. The claimant is Carlos Toledo, an independent researcher, who states that the work was produced with substantial AI assistance from Claude (Anthropic), the system the proof claim names, that they checked the mathematics line by line, and that nothing has been refereed or formalized; the paper describes itself as machine-derived and not peer-reviewed. The proof claim on the site, submitted 2026-10-05, is marked partial by the claimant; the site's label is OPEN, and the problem page's commentary (last edited 8 April 2026) records the two bounds and the conjecture. No outside review, refereed publication or formal proof exists, and this corpus has not rerun the certificates, so no evidence kind is listed. The claim is partial and derives nothing for the problem's standing.