Máté Matolcsi: Upper bounds on the density of planar sets avoiding the unit distance

A planar set A is called 1-avoiding if the distance between any two points of A is not equal to 1. What is the maximal possible density of such a set A? According to an old conjecture of Erdos it is less than 0.25. The best known construction is due to Croft and it has density 0.229... In this talk I will review recent developments of the upper bound getting very close to 0.25. Interestingly, there are two different techniques, giving very similar numerical upper bounds: one using Fourier analysis, and the other using fractional chromatic numbers of unit distance graphs. Yet, neither technique could so far reach the conjectured upper bound of 0.25. Joint work with Gergely Ambrus