A unit-distance graph in the plane with independence ratio below 1/4

Abstract

We prove that there exists a finite unit-distance graph in the plane with independence ratio strictly smaller than 1/4, answering a question of Erdős. Our proof closely follows the framework of Matolcsi, Ruzsa, Varga, and Zsámboki, based on the geometric fractional chromatic number, but adds a carefully chosen two-vertex augmentation that pushes their 27-vertex construction from geometric fractional chromatic number $4$ to a value strictly larger than 4. This disproves their Conjecture 1, and implies that the fractional chromatic number of the plane is strictly larger than 4. The proof can be made fully constructive, but the resulting finite graph has an enormous number of vertices.