On the general no-three-in-line problem
Pith reviewed 2026-05-24 13:26 UTC · model grok-4.3
The pith
One can place at least roughly n to the d-1 times d to the 1 over 2d points in any d-dimensional n-grid with no three collinear.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The number of points that can be placed in the grid n×n×⋯×n (d times) for all d∈N with d≥2 so that no three points are collinear satisfies the lower bound ≫ n^{d-1} d^{1/(2d)}. This extends the result of the no-three-in-line problem to all dimension d≥3.
What carries the argument
A construction or counting argument that produces a point set of the stated size in the d-dimensional integer grid while ensuring no three points lie on a line.
If this is right
- The bound holds for every fixed dimension d at least 2.
- The proportion of the grid occupied by such a set is at least on the order of d to the 1 over 2d divided by n.
- The construction applies equally to the classical d=2 case and all higher dimensions.
- The size grows linearly with the area of any fixed-dimensional slice of the grid.
Where Pith is reading between the lines
- The factor d to the 1 over 2d approaches 1 for large d, suggesting the bound becomes asymptotically n to the d-1.
- Similar counting ideas might apply to avoiding other linear configurations such as four points on a line.
- The result could be tested by direct computation of maximal sets for small values of d and moderate n.
- Extensions to grids with real coordinates or to avoiding collinearity in other metrics remain open.
Load-bearing premise
A construction or counting argument exists which produces a point set of the stated size in every dimension d≥2 while respecting the no-three-collinear condition on the integer lattice.
What would settle it
An exhaustive enumeration for d=3 and n=4 that finds the maximum no-three-collinear set smaller than the claimed lower bound would settle the claim for that case.
read the original abstract
In this paper, we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ so that no three points are collinear satisfies the lower bound \begin{align} \gg n^{d-1}\sqrt[2d]{d}.\nonumber \end{align} This extends the result of the no-three-in-line problem to all dimension $d\geq 3$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that for every dimension d ≥ 2, the maximum size of a subset of the d-dimensional grid [n]^d with no three points collinear satisfies the lower bound ≫ n^{d-1} d^{1/(2d)}, extending the classical no-three-in-line problem from d=2 to higher dimensions.
Significance. If the claimed lower bound were established by a valid construction or counting argument, the result would supply the first explicit quantitative lower bound for the generalized no-three-in-line problem in dimensions d ≥ 3. The bound is consistent with the known 2-D regime (though quantitatively weaker) and does not contradict existing upper-bound literature.
major comments (1)
- The manuscript consists solely of the abstract; no construction, counting argument, or derivation is supplied to support the asserted lower bound. Consequently the central existence claim cannot be verified.
Simulated Author's Rebuttal
We thank the referee for reviewing our manuscript and for the detailed feedback. We address the major comment below.
read point-by-point responses
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Referee: The manuscript consists solely of the abstract; no construction, counting argument, or derivation is supplied to support the asserted lower bound. Consequently the central existence claim cannot be verified.
Authors: We agree that the version under review contains only the abstract statement of the claimed lower bound and does not include any explicit construction, counting argument, or derivation. This prevents verification of the central claim from the submitted text alone. We will revise the manuscript to incorporate the full details of the construction that establishes the bound ≫ n^{d-1} d^{1/(2d)}. revision: yes
Circularity Check
No significant circularity; existence claim with no visible derivation chain
full rationale
The provided abstract states an existence lower bound ≫ n^{d-1} d^{1/(2d)} for no-three-collinear subsets of the d-dimensional grid but contains no equations, constructions, counting arguments, or citations. Without any load-bearing steps, self-definitions, fitted inputs presented as predictions, or self-citation chains, no reduction of the claimed result to its own inputs by construction can be exhibited. The central claim remains an independent existence assertion whose validity depends on an external construction or proof not shown here; this is the normal case for an abstract-only view and yields no circularity.
discussion (0)
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