  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.
