On s-intersecting curves and related problems

Let <i>P</i> be a set of <i>n</i> points in the plane and let <i>C</i> be a family of simple closed curves in the plane each of which avoids the points of <i>P</i>. For every curve <i>C</i> ∈ <i>C</i> we denote by disc<i>(C)</i> the region in the plane bounded by <i>C</i>. Fix an integer <i>s</i> > 0 and assume that every two curves in <i>C</i> intersect at most <i>s</i> times and that for every two curves <i>C</i>,<i>C'</i> ∈ <i>C</i> the intersection disc<i>(C)</i> ∩ disc<i>(C')</i> is a connected set. We consider the family <i>F</i> = {<i>P</i> ∩ disc<i>(C)</i> | <i>C</i> ∈ <i>C</i>}. When <i>s</i> is even, we provide sharp bounds, in terms of <i>n, s,</i> and <i>k</i>, for the number of sets in <i>F</i> of cardinality <i>k</i>, assuming that ∩<sub><i>C ∈C</i></sub>disc<i>(C)</i> is nonempty. In particular, we provide sharp bounds for the number of halving pseudo-parabolas for a set of <i>n</i> points in the plane. Finally, we consider the VC-dimension of <i>F</i> and show that <i>F</i> has VC-dimension at most <i>s+1</i>.

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