5 Colorable Visibility Graphs Have Bounded Size or 4 Collinear Points

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.