Lines and many points on both sides (Q1434251)

This paper considers sets of points \(G\) in the plane that are not contained in one line. A \((k,k)\)-line is a line that has at least \(k\) points of \(G\) in both open halfplanes. The author gives a new upper bound on \(f(k,k)\), the maximum size of a set \(G\) without a \((k,k)\)-line by showing that \(f(k,k)\leq 2k+O(\log\log k)\). This improves previous results by \textit{Y. S. Kupitz} [Combinatorica 13, No. 3, 249--258 (1993; Zbl 0790.52003), J. Comb. Theory, Ser. A 65, No. 1, 117--136 (1994; Zbl 0798.52003)], Alon, and Perles.