Claims

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2019_12_04_mathialagan: Among k points, k at most the cube root of n, some point determines at least a constant times root kn distances to n other points, so alpha_k is at least a constant times root k and the answer is yes; refereed, curator-credited.

2026_01_29_feng: The Aletheia agent (Gemini Deep Think) proves alpha_k at least a constant times the fourth root of k, hence the answer yes, by a Pach–Sharir incidence count over the distance circles of the k poorest points; claimed.

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