Abstract
We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like $$g^{k+1}$$, up to a factor of $$\log g$$. The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are LERF.
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