More on an Erdős-Szekeres-type problem for interior points
As a variant of the well-known Erdős--Szekeres problem, \textit{D. Avis, K. Hosono} and \textit{M. Urabe} [Discrete Math. 241, 33--40 (2001; Zbl 1087.52505)] investigated the following problem: for every integer \(k\geq 1\), determine the smallest integer \(g(k)\), such that every set \(A\) of points in general position in the plane, which has at least \(g(k)\) points in the interior of the convex hull of \(A\), has a subset \(B\), which has exactly \(k\) of its points in the interior of the convex hull of \(B\). They showed \(g(1)=1\), \(g(2)=4\) and \(g(3)\geq 8\). The existence of \(g(k)\) is still open. The paper under review shows \(g(3)=9\).
scientific article; zbMATH DE number 5646260
scientific article