On the Guy-Kelly Conjecture for the No-Three-In-Line Problem

TL;DR

This paper clarifies and corrects a historical error in the heuristic argument related to the Guy-Kelly conjecture for the no-three-in-line problem, providing new insights into the conjecture's upper bounds.

Contribution

It identifies and corrects a previously unnoticed error in the heuristic argument of Guy and Kelly, refining the conjectured upper bound for the problem.

Findings

Error in Guy-Kelly heuristic identified and corrected

Updated upper bound for the no-three-in-line problem established

Recent work by Prellberg aligns with the corrected conjecture

Abstract

We provide details of the error Gabor Ellmann found in 2004 in a heuristic argument of Guy and Kelly on this problem. This led to a correction of their conjectured upper bound for the no-three-in-line problem. However, details of the issue and its correction, including the actual location of the issue, while simple, do not seem to have appeared in the literature previously. That said, very recent work of Prellberg [5] does contain a derivation of the corrected conjectured upper bound.