Mate Matolcsi

A solution to a conjecture of Erdos on planar sets avoiding unit distances

We prove the following old conjecture of Erdos: if A is a measurable set in the plane, such that no two points of A are unit distance away, then the upper density of A is strictly less than 1/4. To put this in context, the best lower bound is given by a construction of Croft, giving density 0.229...
The proof uses a natural modification of the fractional chromatic number of a unit distance graph. Joint work with G. Ambrus, A. Csiszarik, D. Varga and P. Zsamboki.