Bounds for k-diametral point configurations in Euclidean space

A configuration of n points in R^d, repetitions allowed, is k-diametral if any k of its points contain two at distance equal to the diameter of the configuration, and f_k(E^d) denotes the largest possible n. Bezdek and Lángi proved (k−1)(d+1) ≤ f_k(E^d) ≤ (k−1) b(E^d), with equality on the left for d ≤ 3, where b(E^d) is the Borsuk number of finite subsets of R^d, so their upper bound is exponential in d. We prove f_k(E^d) ≤ (d+1) fk−1(E^d) for k ≥ 4, hence f_k(E^d) ≤ 2(d+1)ᵏ⁻², and the subquadratic bound f_4(E^d) ≤ 15 d3/2, by making the Lorentzian Gram matrix of a configuration positive semidefinite through a low rank correction attached to a largest regular simplex. The same method gives f_k(E^d) = O(d(k−1)/2) for every fixed k ≥ 4. We show that f_k(E^d)/(k−1) converges, as k → ∞, to the supremum of the fractional chromatic numbers of the diameter graphs of finite subsets of R^d, and we deduce from the two-distance sets of Bondarenko and of Jenrich and Brouwer that f_6(E^64) ≥ 352 > 5·65 and f_6(E^65) ≥ 416 > 5·66, so the equality f_k(E^d) = (k−1)(d+1) fails already for k = 6. For k = 4 no counterexample is known, and we show that f_4(E^d) = 3d+3 would follow from the inequality Σi≠j (1 − |x_i − x_j|²)² ≤ 2n for 4-diametral configurations of diameter 1, which we conjecture. We prove f_k(E^d) = (k−1)(d+1), with the (k−1)-fold regular simplex as the only extremal configuration, for configurations inscribed in a sphere of radius at most 1/√2 that contain d points pairwise at distance 1. The spherical case is a version, with nonnegative inner products, of a problem of Erdős and Rosenfeld on nearly orthogonal vectors.