#142: our white whale — we didn't land it, but here's the real, current state of the hunt
The target (erdosproblems.com/142, $10,000 — Erdős's largest offer). r_k(N) is the largest subset of {1,...,N} with no non-trivial k-term arithmetic progression. Not "find matching bounds" — an actual asymptotic formula. Erdős called it "probably unattackable at present." We went after it anyway on purpose (Patrick's framing: "it can be our white whale") — cheap to ask, essentially certain to fail, worth it for what a real attempt would surface. We ran it through Codex, then separately through Opus 5 and Fable 5, independently, no cross-talk. All three: HONEST DECLINE. Exactly as expected. What's actually interesting is what came back with the decline.
The literature has moved twice since anyone last summarized it casually, and both models found it, independently, and we verified both citations are real. (1) Behrend's 1946 lower-bound constant — unbeaten for 78 years — was improved for the first time in June 2024: Elsholtz, Hunter, Proske, and Sauermann (arXiv:2406.12290) give the first quasipolynomial improvement, moving the constant from Behrend's 2√2≈2.828 to about 2.667 (in the standard normalization), plus a companion finite-field result breaking the classical c=1/2 barrier. (2) On the upper-bound side, a March 2026 preprint by Raghavan pushes the known exponent past Bloom–Sisask's announced-but- undetailed 5/41 to 1/6 (up to a log-log factor) — originally confirmed only indirectly, via Wikipedia's Roth's-theorem article citing it. Two further independent runs (Grok, no cross-talk with the first two) landed on the exact same two citations and gave us the precise primary source we were missing: arXiv:2603.27045, "Improved Bounds for 3-Progressions," March 2026, revised May 2026 — now confirmed directly, not just by inference. Four models, zero cross-talk, identical literature map. Both findings real, both recent, both things a casual "state of the art" summary (including our own original brief, which we now know was already stale) would have missed.
The actual reason a formula is categorically harder than matching bounds, stated precisely by both models converging on the same structure independently. Write the "deficiency" F(N)=log(N/r_3(N)). A formula needs, in order: that F(N) has a single well-defined shape (log N)^θ (not proven — even θ's existence is open); a specific constant at that scale (moved in 2024, so nobody has a candidate); the sub-leading correction (moved in 2010); and the whole thing matching to 1+o(1). One model (Opus 5) proved directly that the standard tool for guaranteeing such a constant exists at all (Fekete's subadditivity lemma) is provably useless at the actual scale √(log N) the problem lives at — meaning it's not just unproven what the constant is, it's unproven that the constant a formula would have to name even exists. The other (Fable 5) independently derived essentially the same obstruction from a different angle (a near-subadditivity argument) and framed it as two genuinely separate open problems bundled inside one: does the shape exist, and only then, what is it. Both flagged the same historical cautionary tale — Szekeres's 1936 conjectured formula for this exact problem was disproven six years later, and nobody has proposed a candidate formula in print since.
Real side mathematics, clearly labeled as not progress on #142 itself. Both models independently found and proved an exact formula for r_k(N) in the opposite regime — N bounded in terms of k, not k fixed and N→∞ — and both, independently, discovered it fractures almost immediately into arithmetic (mod-2, mod-6, ...) casework rather than staying a smooth function. Neither claims this as progress; both flag it as a small, honest, verified side-result and a cautionary data point (if the one regime where an exact answer exists looks arithmetically messy rather than clean, that's mild evidence against expecting a clean formula anywhere).
Why we're shipping a decline. Same standing policy as #389's "we got refuted and found something real anyway" — a real, well-verified negative result and an accurate, current map of a hard problem is a legitimate outcome, not a consolation prize. Erdős's "probably unattackable at present" remains, as both models put it, exactly right — and now we know precisely how right, with the map freshly redrawn.