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

Paul M Voutier

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.

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

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.
Paper Structure (2 sections, 1 theorem, 8 equations)

This paper contains 2 sections, 1 theorem, 8 equations.

Table of Contents

  1. Introduction
  2. The Heuristic Argument

Key Result

Theorem 1

The number, $t_{n}$, of sets of $3$ collinear points that can be chosen from $S_{n}$ is

Theorems & Definitions (3)

  • Conjecture
  • Theorem
  • Remark