Points defining triangles with distinct circumradii (Q2341939)

Paul Erdős claimed that for any given \((k-1)(k-2){k-1\choose 3}+k\) points in the plane in general position (no 3 on a line and no 4 on a circle), there are some \(k\) points among them having the property that any 3 of them define a different circumradius [\textit{P. Erdős}, Aust. Math. Soc. Gaz. 5, 52--54 (1978; Zbl 0417.52002)]. The paper points to a flaw in the argument (but not a counterexample to the claim) and makes a similar claim that a certain constant times \(k^9\) points suffices.