2013-12 Equilateral triangle in R^n

Let \( A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n \). Prove that, for any \( X \subset A \) with \( |X| > 2^{n+1}/n \), there exist three distinct points in \( X \) that are the vertices of an equilateral triangle.

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2 thoughts on “2013-12 Equilateral triangle in R^n

  1. student

    $Latex A = \{(a_1, a_2, \cdots, a_n) : a_i = \pm 1 (i = 1, 2, \cdots, n)\}$ 인가요?

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