Claims

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1960_01_01_erdos: For d at least 4 and p the floor of d/2, the maximum number of unit distances among n points of d-space is ((p-1)/(2p) + o(1)) n^2: Lenz's construction from below and the Erdős–Stone theorem from above.

1967_01_01_erdos: For even d at least 4 and n large, the maximum number of unit distances among n points of d-space equals the Turán number t_p(n) plus n whenever 2d divides n and lies within p of it otherwise; p is d/2.

1990_09_01_erdos_pach: For odd d at least 5 and p the floor of d/2, the maximum number of unit distances among n points of d-space is (p-1)/(2p) n^2 plus a second-order term of exact order n^(4/3).

1997_01_01_brass: For n at least 5, the maximum number of unit distances among n points of four-dimensional space is the floor of n^2/4 plus n when 8 or 10 divides n, and one less otherwise; completed by a number-theoretic result of van Wamelen.

2007_07_02_swanepoel: For d at least 4 and n large in terms of d, the n-point sets of d-space with the most unit distances are Lenz configurations, which gives the exact value of f_d(n) for every even d at least 6.

2026_09_23_openai: Theorem 1.1 of the OpenAI release manuscript of 23 September 2026 proves that n points in the plane determine at most C n^beta unit distances for absolute C and beta < 4/3; accepted on built Lean as a partial claim, the plane only.

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