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Claim. In A sunflower anti-Ramsey theorem and its applications, Corollary 1(1) states that for every dimension $d$ there is a constant $c_d$ such that, if $X\subset\mathbb{R}^d$ has $|X|\ge c_dn^{2d+1}/\log n$ and no $d+2$ points of $X$ lie on a $(d-1)$-sphere, then $X$ contains $n$ points all of whose $d$-simplices have distinct circumradii. The proof colors each $(d+1)$-tuple by its circumradius and applies the paper's sunflower anti-Ramsey theorem with $\lambda=2$, since at most two spheres of a given radius pass through $d$ points. For $d=2$ this gives $n_k=O(k^5/\log k)$ for any planar point set with no four points concyclic.

Covers. The upper bound $n_k=O(k^5/\log k)$. Its hypothesis, no four points on a circle, is implied by the general position of Problem 827, so the bound holds for the problem's $n_k$. It improves the $O(k^9)$ bound of Martínez and Roldán-Pensado and does not determine $n_k$ for any $k$.

Standing. Claimed. The manuscript is on arXiv, dated 19 May 2015, and no journal version of it is recorded. A thread post of 26 August 2026 pointed to Corollary 1(1) as giving this bound. The site's commentary does not mention the manuscript, and no review of it is recorded.