What does this research mean for the field?

Independent computational methods confirm the published values and optimal configuration counts for the maximum number of non-coplanar points in an n×n×n grid (OEIS A280537) up to n=5, and provide explicit witnesses for a(6) ≥ 16 and a(7) ≥ 18. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

To compute and rigorously verify bounds for OEIS A280537, which tracks the maximum number of points in an n×n×n grid containing no four coplanar points.

August 23, 2026Open Access

No four coplanar in the cube: a witness for a(7) >= 18 and the state of the upper bound (OEIS A280537)

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Authors

AKAleksei KudriashovNational Heritage Institute

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Implication

Computational study reveals explicit witnesses for a(7) >= 18 in three-dimensional grids, confirming exact configuration counts for smaller cubes while leaving upper bounds open.

Key Points

To compute and rigorously verify bounds for OEIS A280537, which tracks the maximum number of points in an n×n×n grid containing no four coplanar points.
Implemented three independent computational enumeration and SAT-solving pipelines with symmetry pruning.
Generated and externally validated DRAT proof certificates for unsatisfiability, specifically verifying the impossibility of 14 points in a 5×5×5 cube.
Reproduced exact optimal counts for small grids: 1 configuration class for n=3 (size 8), 232 classes for n=4 (size 10), and 38 classes for n=5 (size 13).
Identified 4 explicit witness configurations establishing a(6) >= 16 and 6 witness configurations establishing a(7) >= 18.

Cite This Study

Aleksei Kudriashov (2026) studied this question.

synapsesocial.com/papers/6a8aae207677a34114446f25 https://doi.org/10.5281/zenodo.22023080

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Implication

Computational study reveals explicit witnesses for a(7) >= 18 in three-dimensional grids, confirming exact configuration counts for smaller cubes while leaving upper bounds open.

No four coplanar in the cube: a witness for a(7) >= 18 and the state of the upper bound (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

No four coplanar in the cube: a witness for a(7) >= 18 and the state of the upper bound (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

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