Incidence lower bounds and applications
Incidence lower bounds and applications

Alex Cohen, MIT
Fine Hall 214
*Special Discrete Math Seminar hosted by PACM*
Lots of problems in combinatorics and analysis are connected to upper bounds for incidences: given a set of points and tubes, how much can they intersect? We prove that if you choose ʽnʼ points and a line through each point, there is a nontrivial pointline pair with distance <= n^{2/3+o(1)}. It quickly follows that in any set of ʽnʼ points in the unit square some three form a triangle of area <= n^{7/6+o(1)}, a new bound for this problem. The main work is proving a more general incidence lower bound result under a new regularity condition.
Joint with Cosmin Pohoata and Dimitrii Zakharov.