  We prove that on a punctured oriented surface with Euler characteristic chi <
0, the maximal cardinality of a set of essential simple arcs that are pairwise
non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a
cubic estimate in |chi| for a set of curves pairwise intersecting at most once
on a closed surface. We also give polynomial estimates in |chi| for sets of
arcs and curves pairwise intersecting a uniformly bounded number of times.
Finally, we prove that on a punctured sphere the maximal cardinality of a set
of arcs starting and ending at specified punctures and pairwise intersecting at
most once is 1/2|chi|(|chi|+1).
