On the number of ordinary circles determined by \(n\) points (Q635749)

For a given set \(B\), an ordinary line (respectively, an ordinary circle) is one that passes through exactly two (respectively, three) points of \(B\). It is proved by \textit{L. M. Kelly} and \textit{W. O. J. Moser} in [Can. J. Math. 10, 210--219 (1958; Zbl 0081.15103)] that if \(B\) consists of \(n\) non-collinear points, then the number of ordinary lines that \(B\) admits is at least \(\frac{3n}{7}\), and this result is strengthened by \textit{J. Csima} and \textit{E. T. Sawyer} in [Discrete Comput. Geom. 9, No. 2, 187--202 (1993; Zbl 0771.52003)]. This lavishly answers a modest question of Sylvester asking whether there is at least \textit{one} such line. As for circles, it is proved by \textit{P. D. T. A. Elliott} in [Acta Math. Acad. Sci. Hung. 18, 181--188 (1967; Zbl 0163.14701)] that if \(B\) consists of \(n\) non-collinear non-concyclic points, then the number of ordinary circles that \(B\) admits is at least \(\frac{4}{63}{n\choose2}\), and this lower bound is increased to \(\frac{22}{247}{n\choose2}\) by \textit{A. Bálintova} and \textit{V. Bálint} in [Acta Math. Hung. 63, No. 3, 283--289 (1994; Zbl 0796.51008)]. The paper under review raises this lower bound to \(\frac{1}{9}{n\choose2}\). Its introduction gives a nice brief history. It is often felt that lines are a special kind of circles. Thus calling (ordinary) lines and circles (ordinary) general circles, and considering sets consisting of \(n\) points that do not all lie on a general circle, this reviewer raises the question about a fairly sharp lower bound for the number of ordinary general circles that such sets admit. One may also ask similar questions regarding lines (respectively circles) that pass through \textit{at least} (rather than exactly) two (respectively three) points of the given set. Some cases of these questions are treated in Chapter 10 of [\textit{M. Aigner} and \textit{G. M. Ziegler}. Proofs from THE BOOK. 4th revised and enlarged ed. Berlin: Springer (2010; Zbl 1185.00001)].