The largest isosceles triangle-free subset of the grid
Adam Wagner, Worcester Polytechnic Institute
Abstract: How many points can we pick in the N by N grid, without choosing three points that satisfy d(a,b) = d(b,c), i.e. a (possibly flat) isosceles triangle? This beautiful question was recently asked by Jordan Ellenberg. I will start by discussing the little we managed to prove about the asymptotics of this function. In the second half of the talk, we will look at the structure of the optimal constructions for small values of N. The best known constructions, found by computer searches for small values of N, clearly follow a pattern which we do not yet understand. We will discuss how one can train transformers to understand this pattern, and use this trained transformer to help us find a bit better constructions for various N.
This is joint work with Jordan Ellenberg, Marijn Heule, Yuval Wigderson, and Geordie Williamson.
