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

TL;DR

This paper determines the maximum number of points that can be chosen from an n-by-n grid without having k+1 collinear points, proving it equals kn under certain conditions on k.

Contribution

It establishes the exact maximum for large k relative to n, resolving a problem open for over a century, using novel probabilistic graph methods.

Findings

Maximum points without k+1 collinear points is kn for large k

Proves the result for k > C√(n log n)

Uses bi-uniform random bipartite graphs and concentration inequalities

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.