Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results

2026-09-22

AI Summary

研究解决了平面点集中寻找共线点或几乎完全图的问题，通过证明大平面点集要么包含四个共线点，要么包含七个最多一对不可见点的点。

Abstract

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

Problem

• planar point set
• collinear points
• visibility graph
• chromatic number

Innovation

• planar point set
• collinear points
• visibility graph
• chromatic number
• big-line-big-clique conjecture