Swanepoel 2009 unit distances diameters euclidean spaces

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corollary_2: Swanepoel's exact formula for the maximum number u_d(n) of unit distances among n points of R^d, for every even d >= 6 and all n sufficiently large in terms of d, as the Turán number t_p(n) plus a term fixed by n modulo 2d.

corollary_3: Swanepoel's exact formulas for the maximum number M_d(n) of diameters among n points of R^d, separately for d = 4, d = 5, even d >= 6 and odd d >= 7, for all n sufficiently large in terms of d.

corollary_6: Swanepoel's corollary of the stability theorems: for d >= 4, an n-point set in R^d with ((p-1)/2p - o(1))n^2 unit distance pairs is a Lenz configuration apart from o(n) points.

definition_p2: Swanepoel's definitions of the maximum numbers u_d(n) of unit distances and M_d(n) of diameters among n points of R^d, of Lenz configurations in every dimension d >= 4, and of extremal sets.

theorem_1: Swanepoel's main theorem: for every d >= 4 there is N(d) such that every set of n >= N(d) points in R^d with the most unit distances, or with the most diameters, is a Lenz configuration.

theorem_4: Swanepoel's even-dimensional stability theorem: an n-point set in R^d, d >= 4 even, with nearly the Lenz number of unit distances splits into a small exceptional part and p = d/2 nearly equal parts on mutually orthogonal concentric circles.

theorem_5: Swanepoel's odd-dimensional stability theorem: an n-point set in R^d, d >= 5 odd, with nearly the Lenz number of unit distances splits into a small exceptional part, a part on a 2-sphere and p - 1 parts on circles, all concentric and mutually orthogonal.

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Swanepoel, Konrad J., Unit distances and diameters in Euclidean spaces. Discrete Comput. Geom. 41 (2009), no. 1, 1--27. https://doi.org/10.1007/s00454-008-9082-x

The paper proves that for every d >= 4 and all sufficiently large n (depending on d) the n-point sets in R^d maximizing the number of unit distances, and likewise those maximizing the number of diameters, must be specific types of Lenz constructions -- points placed on concentric circles in pairwise orthogonal planes (with one circle replaced by a 2-sphere in a 3-dimensional subspace when d is odd) of radii r_i satisfying r_i^2 + r_j^2 = 1 for i != j (definition, pp. 2-3; Theorem 1, p. 3). As corollaries the author determines the exact value of u_d(n) for all even d >= 6 (Corollary 2) and the exact value of M_d(n) for all d >= 4 (Corollary 3), again for n large in terms of d. The argument is a stability analysis of the Lenz configuration (Theorems 4 and 5 and Corollary 6, p. 4) combined with extremal graph theory, refining the Erdos-Stone asymptotics u_d(n) = ((p-1)/2p)n^2 + o(n^2) with p = floor(d/2) that Erdos obtained from the absence of K_{p+1}(3) as a unit-distance graph. The introduction surveys the state of the art: u_2(n) = O(n^{4/3}) by Spencer, Szemeredi and Trotter with Erdos's n^{1+c/log log n} lower bound, the d = 3 gap between cn^{4/3} log log n and cn^{3/2}beta(n), Brass and Van Wamelen's exact u_4(n), and Erdos-Pach's u_d(n) = ((p-1)/2p)n^2 + Theta(n^{4/3}) for odd d >= 5. For problems 223 and 1085 the paper determines, for all sufficiently large n, the exact maximum number of diameters for every d >= 4 and the exact maximum number of unit distances for every even d >= 6, and shows that the extremal sets are Lenz configurations. It does not determine the exact unit-distance count for odd d >= 5, where for large n it would follow from the maximum number of unit distances among m points on a 2-sphere, of radius 1/sqrt(2) for d >= 7 and of arbitrary radius for d = 5 (pp. 3, 11 and 14); for d = 4 it cites Brass and Van Wamelen, and d = 2 and d = 3 are not its subject.

Source: https://arxiv.org/abs/0707.0213. The copy read for this card is arXiv:0707.0213v1 (2 July 2007), 24 pages; the theorem labels and page numbers below are that preprint's. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:0707.0213), every other right reserved.

Bears on. #223: the problem's f_d(n) is the paper's M_d(n), the most diameters among n points of R^d, and Corollary 3 (p. 4) gives its exact value for every d >= 4 and every n sufficiently large in terms of d, with Theorem 1 (p. 3) describing the extremal sets as Lenz configurations; nothing is proved for d = 2, d = 3 or smaller n. #1085: the problem's f_d(n) is the paper's u_d(n); Corollary 2 (p. 3) gives its exact value for every even d >= 6 and every n sufficiently large in terms of d, and Theorem 1 shows for every d >= 4 and large n that the extremal sets are Lenz configurations, which for odd d >= 5 leaves the exact value depending on the maximum number of unit distances among points on a 2-sphere, of radius 1/sqrt(2) for d >= 7 and of arbitrary radius for d = 5 (pp. 3, 11 and 14); the exact value for d = 4 is cited from Brass and Van Wamelen (p. 2), and nothing is proved for d = 2 or d = 3.

Results. Labels and pages are those of arXiv:0707.0213v1.

• Definitions (pp. 1-3): u_d(n), M_d(n), Lenz configurations for even and odd d >= 4, and extremal sets.
• Theorem 1 (p. 3): for d >= 4 and n >= N(d), extremal sets for unit distances or diameters are Lenz configurations.
• Corollary 2 (p. 3): the exact value of u_d(n) for even d >= 6 and large n.
• Corollary 3 (p. 4): the exact value of M_d(n) for every d >= 4 and large n.
• Theorem 4 (p. 4): stability for even d >= 4, near-extremal sets lie mostly on orthogonal concentric circles.
• Theorem 5 (p. 4): stability for odd d >= 5, near-extremal sets lie mostly on a 2-sphere and orthogonal concentric circles.
• Corollary 6 (p. 4): a set with ((p-1)/2p - o(1))n^2 unit distances is a Lenz configuration except for o(n) points.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.