Abstract
How large can a set of simple closed curves on a torus be, such that any two curves are non-homotopic and intersect at most k times? It is known since 1996 that for any fixed k, such a set must be finite. The topic has been extensively studied, leading to a recent upper bound of k + O(k^½ log k) on the size of the set, established by Aougab and Gaster.
We resolve the problem by determining the optimal bound and providing a matching construction for every value of k. In particular, we show that the size of such a set never exceeds k + 6, and is at most k + 4 for sufficiently large k.
In this talk, we will present the main ideas behind the proof, which utilizes well-known tools from combinatorics, discrete optimization, and geometry, along with some number-theoretic observations.