The no-four-on-circle problem
Let \(p\) be a prime number. Define as a subset of the \(p\times p\)-grid the point set \[ P(p)= \{(t, t^2\text{ mod } p)\mid 0\leq t< p/4\}. \] Calculating determinants, the author shows that there are no four different points in \(P(p)\) on a common circle, and no three different points in \(P(p)\) on a common line. As a consequence, let \(C(n)\) denote the number of \(n\times n\)-grid points, no four of them on a common line, and let \(\varepsilon> 0\) be a real constant. Then, for sufficiently large \(n\), \(C(n)\geq ({1\over 4}- \varepsilon)n\).
scientific article; zbMATH DE number 802362