ADJUSTING A CONJECTURE OF ERDOS

We investigate a conjecture of Paul Erdos, 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 are unit vectors in a Hilbert space, then at least C2(n)/n of all epsilon is an element of {-1, 1}(n) are such that vertical bar Sigma(n)(i=1) epsilon(i)v(i) vertical bar <= 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 Erdos. We prove some weaker related results that shed some light on the hardness of the problem.