Erdős Problem #160

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Erdős asked how many colours [N] needs so that every four-term progression sees at least three of them. The page records N^{log 3/log 22+o(1)} ≈ N^{0.355} above and exp(c(log N)^{1/9}) below. The proof-claims tab now carries two preprints: Rio Itabe (14 July 2026) with N^{1/3+o(1)} and a Lean 4 development, and Shi–Dong (arXiv 2607.20752, 22 July) with N^{1/4+o(1)} — both from the same new idea, a base-p carry control plus the cubic field norm F_{p³} → F_p, which handles the one two-colour pattern (ABBA) that Behrend-style colourings are not known to. We read both proofs lemma by lemma, re-implemented Shi–Dong's colouring twice (norm as z^{1+p+p²} in Python, as a determinant in C) and enumerated every 4-AP in ℤ/p⁴ℤ for p = 5, 7, 11 (and in C for p = 13) — none symmetrically coloured, palette within 27p, three controls fire — sampled their k = 6 colouring of ℤ/7⁹ℤ (2.2·10⁸ progressions, none symmetric), brute-forced Itabe's three labels, kernel-replayed his Lean (standard axioms, statement matched to the page's h), and sent four hostile Opus seats at the papers. All four seats and the host read agree: nothing fatal, no gap that reaches a theorem; two one-line gaps in Shi–Dong's written proofs (a step-exactly-p carry case Lemma 2.1 does not cover; a false reason at x = M/2 in Lemma 5.3, true conclusion), a stale 'strongest previously known' sentence, and, on priority, that Itabe's manuscript was public eight days before Shi–Dong's v1, which did not cite it; v2 cites it as 'a concurrent result'. Itabe's cubic factor is, up to the choice of basis, their k = 4 colouring restricted to pℤ/p⁴ℤ. Ours: the referee report, the checks, and exact values we could not find written down anywhere — h(N) = 3, 4, 5, 6 on 4–12, 13–22, 23–35, 36–51. The problem stays open.