We investigate a conjecture of Paul Erdős, 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 v1, …, vn are unit vectors in a Hilbert space, then at least C2nn of all ϵ∈−1, 1n are such that |∑ni=1ϵivi|≤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 Erdős. We prove some weaker related results that shed some light on the hardness of the problem.
Carnielli et al. (Tue,) studied this question.
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