Rank bound for equidistant point configurations

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Problem 7.1.4 from Глибичук А.А., Дайняк А.Б., Ильинский Д.Г., Купавский А.Б., Райгородский А.М., Скопенков А.Б., Чернов А.А. "Элементы дискретной математики в задачах" [ГДИ+16]More problems in Section 7.1 "The Linear-Algebraic Method in Combinatorics"

The distance between points of the space Rn\mathbb {R}^{n} is defined by the formula

∣(x1,…,xn),(y1,…,yn)∣:=(x1−y1)2+…+(xn−yn)2. \left|(x_{1}, \ldots , x_{n}), (y_{1}, \ldots , y_{n})\right| := \sqrt{(x_{1}-y_{1})^{2} + \ldots + (x_{n}-y_{n})^{2}}.
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(1)

The largest number of points in Rn\mathbb {R}^{n} with equal pairwise distances is n+1n+1.

(2)

Construct n(n−1)2\dfrac {n(n-1)}{2} points in Rn\mathbb {R}^{n} whose pairwise distances take only two distinct values.

(3)

For a∈Ra \in \mathbb {R} and points v=(v1,…,vn)∈Rnv = (v_{1}, \ldots , v_{n}) \in \mathbb {R}^{n}, x=(x1,…,xn)∈Rnx = (x_{1}, \ldots , x_{n}) \in \mathbb {R}^{n}, denote Pv(x):=∣x−v∣2−a2P_{v}(x) := \left|x-v\right|^{2} - a^{2}. If the pairwise distances between kk points u1,…,uk∈Rnu_{1}, \ldots , u_{k} \in \mathbb {R}^{n} are equal to aa, then the polynomials Pu1,…,PukP_{u_{1}}, \ldots , P_{u_{k}} are linearly independent over Q\mathbb {Q}.

(4)

If the pairwise distances between kk points in Rn\mathbb {R}^{n} take only two distinct values, then k≤(n+1)(n+4)2k \leq \dfrac {(n+1)(n+4)}{2}.

References

[ГДИ+16] Глибичук А.А., Дайняк А.Б., Ильинский Д.Г., Купавский А.Б., Райгородский А.М., Скопенков А.Б., Чернов А.А. Элементы дискретной математики в задачах. МЦНМО, 2016.