For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.
Large grid subsets without many cospherical points
2 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the previously best known lower bound of $n/4$ toward the Erd\H{o}s--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $\Omega \bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $\Omega \bigl( n^{\frac{d}{d+1}} \bigr)$ bound in a strong sense, and asymptotically resolves the generalized Erd\H{o}s--Purdy problem posed by Brass, Moser, and Pach.
years
2026 2representative citing papers
Geometry-aware MCTS with incremental constraint updates and symmetry pruning yields new best-known configurations for five of six tested combinatorial geometry problems, including ~1.8n points for Max-N3IL on grids 82-119.
citing papers explorer
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No-$(k+1)$-in-line problem for $k \geqslant 3$
For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.
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Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry
Geometry-aware MCTS with incremental constraint updates and symmetry pruning yields new best-known configurations for five of six tested combinatorial geometry problems, including ~1.8n points for Max-N3IL on grids 82-119.