Arcs intersecting at most once

We prove that on a punctured oriented surface with Euler characteristic χ < 0 , the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|χ|(|χ| + 1) . This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |χ| 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|χ|(|χ| + 1) .