On curves intersecting at most once

TL;DR

This paper proves an upper bound on the number of simple closed curves on a surface that intersect at most once, matching known constructions up to a logarithmic factor, using probabilistic graph theory methods.

Contribution

It establishes a near-optimal bound on the maximum number of such curves, extending to curves intersecting at most k times, with a novel probabilistic proof technique.

Findings

Bound of approximately g^2 log(g) for curves intersecting at most once

Extension of the bound to curves intersecting at most k times

Use of probabilistic graph theory in topological curve analysis

Abstract

We prove that on a closed surface of genus $g$, the cardinality of a set of simple closed curves in which any two are non-homotopic and intersect at most once is $\lesssim g^2 \log(g)$. This bound matches the largest known constructions to within a logarithmic factor. The proof uses a probabilistic argument in graph theory. It generalizes as well to the case of curves that intersect at most $k$ times in pairs.