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An improved lower bound for an Erdős-Szekeres-type problem with interior points. (English) Zbl 1268.52012

Summary: For each finite planar point set \(P\) with no three collinear points, an interior point of \(P\) is a point in \(P\) such that it is not on the boundary of the convex hull of \(P\). For any integer \(k > 0\), let \(g(k)\) be the smallest integer such that every finite planar point set \(P\) with no three collinear points and with at least \(g(k)\) interior points has a subset \(Q\) whose the interior of the convex hull of \(Q\) contains exactly \(k\) points of \(P\). In this paper, we show that \(g(k) \geq k^2\) for all integer \(k\geq 4\).

MSC:

52C10 Erdős problems and related topics of discrete geometry
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