Abstract
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 \({\frac{1}{2}|\chi|(|\chi| + 1)}\).
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P. Przytycki was partially supported by National Science Centre DEC-2012/06/A/ST1/00259 and NSERC.
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Przytycki, P. Arcs intersecting at most once. Geom. Funct. Anal. 25, 658–670 (2015). https://doi.org/10.1007/s00039-015-0320-0
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DOI: https://doi.org/10.1007/s00039-015-0320-0