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August 17, 2025

Adjusting a conjecture of Erdős

Key Points

The conjecture by Erdős is disapproved, showing a counterexample for hilbert spaces of dimension greater than 2.
For dimension 2, only a small modification is required, indicating the problem's continued complexity.
We present weaker related results that highlight the challenges in fully resolving the conjecture.
The study emphasizes the need for further exploration given the significant mathematical implications.

Abstract

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.

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