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Adjusting a conjecture of Erd s

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Abstract:

We investigate a conjecture of Paul Erd s, the last unsolved problem among those proposed in his landmark paper [2]. The conjecture states that there exists an absolute constant $C > 0$ such that, if $v_1, dots, v_n$ are unit vectors in a Hilbert space, then at least $C frac{2n}{n}$ of all $epsilon in {-1,1}^n$ are such that $|sum_{i=1}^n epsilon_i v_i| leq 1$. We disprove the conjecture. For Hilbert spaces of dimension $d > 2,$ the counterexample is quite strong, and implies that a substantial weakening of the conjecture is necessary. However, for $d = 2,$ only a minor modification is necessary, and it seems to us that it remains a hard problem, worthy of Erd s. We prove some weaker related results that shed some light on the hardness of the problem.

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