Erdős Problem #188
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The set of numbers \(k\) such that \(\mathbb{R}^2\) can be red/blue coloured with no pair of red points unit distance apart, and no \(k\)-term arithmetic progression of blue points with distance 1.
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Colour the plane with no red unit pair; how long a blue unit-spaced progression can be avoided? The page says 6 ≤ k ≤ 10⁷; the forum knows Currier–Mody–Xie–Zhang's June 2026 proof of k ≤ 6330. We recomputed every constant of that proof (hexagonal cells of diameter 1, |z(D)| = 18, p = 1/19, Janson exponent 0.015571, Milnor–Thom 2⁴·5¹⁶·m⁸): it holds, and parametrising the progression by its three real degrees of freedom instead of four gives k ≤ 4917 from the identical argument. Both known lower bounds — EGMRSS's E² → (ℓ₂, ℓ₄) and Tsaturian's E² → (ℓ₂, ℓ₅) — become exact finite certificates: 15 and 69 points of the unit triangular lattice and its Moser-angle rotations, with a red origin and (for ℓ₅) one WLOG blue point, each clause verified exactly in ℚ(√3, √11), minimal, closed by three solvers and drat-trim. The same apparatus at ℓ₆ found no certificate and refuted nothing: the instances that could decide the matched question are thousands of times harder and ran out their time caps undecided (one radius-3 instance found an ℓ₆-free patch in 5,439 s; nothing refuted the route).
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