No-$(k+1)$-in-line problem for large constant $k$

TL;DR

This work resolves the asymptotic regime for the no-$(k+1)$-in-line problem on an $n\times n$ grid by proving that for $n\ge k\ge 10^{37}$ the maximum number of points is exactly $kn$. The authors develop a robust probabilistic framework based on iterative subsampling, the Lovász Local Lemma, and spread distributions to achieve near-regularity on horizontal and vertical lines while maintaining control on all other lines, then extract a configuration with exactly $k$ points per axis line via a Hall-type argument. They extend the methodology to higher dimensions, obtaining near-optimal bounds for $f_{k,d,t}(n)$, and provide a detailed route from the relaxed version to the main no-$k+1$-in-line result. These results advance incidence-geometry bounds in a purely probabilistic setting and demonstrate techniques that may apply to related combinatorial design problems.

Abstract

How many points can be placed in an $n\times n$ grid so that every (affine) line contains at most $k$ points? We prove that for $n \ge k \ge 10^{37}$ the maximum number of points is exactly $kn$. Our proof builds on the recent work of Kovács, Nagy, and Szabó (who proved an analogous result when $k$ is at least about $\sqrt{n \log n}$), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher-dimensional extensions of this problem.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2: jain-pham-24
• Proposition 2.3
• proof
• Proposition 2.5
• proof