5 Colorable Visibility Graphs Have Bounded Size or 4 Collinear Points

TL;DR

This paper explores bounds on the size of colorable visibility graphs with limited collinear points, proving that large point sets must contain either four collinear points or an uncolorable visibility graph.

Contribution

It establishes a specific size bound (2311 points) beyond which point sets must have four collinear points or non-5-colorable visibility graphs, advancing understanding of the Big Line - Big Clique conjecture.

Findings

Any set with at least 2311 points has either 4 collinear points or a visibility graph not 5-colorable.

The result provides a concrete size threshold related to visibility graph colorability.

Supports the conjecture by linking large point sets to geometric and coloring properties.

Abstract

We investigate the question of finding a bound for the size of a $\chi$-colorable finite visibility graph that has at most $\ell$ collinear points. This can be regarded as a relaxed version of the Big Line - Big Clique conjecture. We prove that any finite point set that has at least 2311 points has either 4 collinear points or a visibility graph that cannot be 5-colored.