Certified exact values for the no-three-in-line problem in the three-dimensional grid
Year: 2026
Type: preprint
Abstract: a(n) is the maximum number of points in the n×n×n grid with no three of them collinear in R³. Pór and Wood proved a(n) = Θ(n²); no exact values appear to have been recorded previously. We determine a(1..6) = 1, 8, 16, 28, 40, 64, give certified lower bounds a(7) ≥ 73 and a(8) ≥ 93, and prove a(p) ≥ p² for every prime p by an elementary construction from an anisotropic binary quadratic form. New in version 1.2. The certified route decides satisfiability with a DRAT certificate an independent checker verifies, but it needs one link no certificate can attest: that the propositional formula expresses the problem. Version 1.2 removes that link for small n rather than merely checking it. A plain exhaustive traversal — points placed in the cube, triples tested by integer cross product, no formula, no solver, no certificate — reproduces a(1) = 1, a(2) = 8 and a(3) = 16 from a bar of zero, in 2, 37 and 49812 nodes, so no value could be fed in and echoed back. At n = 4 the same traversal reaches 28 points but does not exhaust its tree within an hour (8.85·10&sup9; nodes), so for n ≥ 4 it supplies the lower bound only and the upper bounds there remain the certified ones. The construction a(p) ≥ p² was additionally built explicitly for p = 3, 5, 7, 11, 13 with every triple tested — 790244 triples at p = 13 — as a guard against a slip in transcription rather than in the proof. Correction to the version 1.1 record. Its description stated a(7) ≥ 71 and a(8) ≥ 89 while the PDF attached to the same record stated 73 and 93. The values are 73 and 93; the description was stale. The note also records a structural anomaly we cannot explain: requiring both outer layers to attain the planar maximum 2n costs nothing at n = 4 and n = 6 but exactly one point at n = 5. Disclosure: the computations, the programs and the text were produced by AI agents (Anthropic Claude) under the direction of the author of record, who posed the question, chose what to compute, decided what to claim and is responsible for the content. Two agents worked independently, one searching and one verifying without reading the searcher's code. Version 1.4. The count of optima at n = 4 is now established by this project as well: a 32-shard traversal, every shard run to exhaustion (13 215 017 061 nodes in total), finds exactly 14 optima of size 28, forming three classes under the full cube group with stabiliser orders 8, 8, 24 (6+6+2 = 14), in agreement with the external recount quoted in v1.3. All fourteen configurations ship in the package. Every optimum at n ≤ 4 has the layer profile [2n, (2n-2)^(n-2), 2n], so the empirical observation is a fact up to n = 4. v1.5 (3 Sep 2026): new certified lower bounds a(8) ≥ 94 (previously 93), a(9) ≥ 116, a(10) ≥ 138, a(11) ≥ 164 from a sweep over all 33 conjugacy classes of subgroups of the cube group (exact optimisation inside each class; the order-12 class is solved to optimality at n = 9 and n = 10); every configuration verified by two independent programs, rigid, and certified locally optimal under exchanges of up to four (73, 94, 116) or three (138, 164) points. Section 5 rewritten accordingly; the version-1.4 account of n = 7, 8 is kept as data. Witnesses and per-class results are archived in doi:10.5281/zenodo.22271375.
Author: Kudriashov, Aleksei
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203739000
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