Combinatorics · Oct 6, 2026

The higher-dimensional Erdős distinct-distances conjecture

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least cdn2/dc_dn^{2/d} distinct distances, with cd>0c_d\gt 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

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