### Máté Matolcsi and Dániel Varga: The density of planar sets avoiding unit distances

Matolcsi Máté#### BBC+G Seminar

on 9/23/22

We prove that the number of tangencies between the members of two families, each of which consists of pairwise disjoint curves, can be as large as . From a conjecture about - matrices it would follow that if the families are doubly-grounded then this bound is sharp. We also show that if the curves are required to be -monotone, then the maximum number of tangencies is , which improves a result by Pach, Suk, and Treml. Joint work with Dömötör Pálvölgyi.