Erdős Problem #654

Let \(f(n)\) be such that, given any \(x_1,\ldots,x_n\in \mathbb{R}^2\) with no four points on a circle, there exists some \(x_i\) with at least \(f(n)\) many distinct distances to the other \(x_j\). Is it true that \(f(n) > (1-o(1))n\)? Erdős asks this in [Er97e]. The answer is no: Aletheia [Fe26] constructed, for every \(m \geq 10\), a set of \(n = 4m\) points with no four on a circle such that every point has fewer than \(\frac{3}{4}n\) distinct distances to the other points. This gives infinitely many \(n\), which is enough to refute the statement with \(\varepsilon = 1/8\). See `Erdos654.erdos_654.variants.feng`.