Triangle colorings require at least seven colors

TL;DR

This paper proves that any coloring of the plane with certain geometric constraints, such as partitioning into triangles or polygons with specific adjacency properties, requires at least seven colors to ensure no two points at distance one share the same color.

Contribution

It establishes a lower bound of seven colors for plane colorings under specific geometric and adjacency conditions, extending previous results in geometric coloring.

Findings

Colorings with triangle partitions need at least seven colors.

Uniformly colored polygons with a point bordering four polygons also require at least seven colors.

The result generalizes known bounds for plane colorings with geometric constraints.

Abstract

We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven colors. This is also true for a coloring using uniformly colored polygons if it has a point bordering at least four polygons.