Settling the no-$(k+1)$-in-line problem when $k$ is not small

TL;DR

The paper addresses the maximal size $f_k(n)$ of a subset of the $n\times n$ grid with no $k+1$ collinear points. It proves that, for sufficiently large $n$ and $k \ge C\sqrt{n\log n}$, the exact value is $f_k(n)=kn$, achieved via a bi-uniform random bipartite-graph construction and concentration arguments, complemented by explicit large-$k$ and divisibility-adjustment techniques. The approach combines probabilistic methods (random $k$-regular subgraphs with two edge-probability regimes), geometric line-counting bounds, and matching-containment counts to control the behavior on all lines. This yields a wide-range regime where the natural pigeonhole upper bound is tight and provides a framework for tackling other grid-extremal problems with line constraints. The results open avenues for refining constants, extending to broader ranges of $k$, and exploring sandwich-type methods in regular bipartite graph settings.

Abstract

What is the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are in a line? This has been asked more than $100$ years ago for $k=2$ and it remained wide open ever since. In this paper, we prove the precise answer is $kn$, provided that $k>C\sqrt{n\log{n}}$ for an absolute constant $C$. The proof relies on carefully constructed bi-uniform random bipartite graphs and concentration inequalities.

Figures (2)

• Figure 1: The complement of a suitable construction for $n=16, k=11$.
• Figure 2: The arrangement of the sparse and dense subgrids and the critical secant lines

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Chernoff bound for the binomial distribution
• Theorem 2.2: Linial-Luria linial2014chernoff
• Theorem 2.3: Erdős
• Theorem 2.4: Special case of isaev2018complex
• proof
• Proposition 3.1
• proof