Adjusting a conjecture of Erdős

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 C 2 n of all ∈ {−1, 1} are such that | Pn i=1 ivi |≤ 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.