ADJUSTING A CONJECTURE OF ERDOS

Author(s)

Carnielli, Walter ^[1] ; Carolino, Pietro K. ^[2]

Affiliation

^[1] State Univ Campinas UNICAMP, Ctr Log Epistemol & Hist Sci, Campinas, SP - Brazil

^[2] State Univ Campinas UNICAMP, Dept Philosophy, Campinas, SP - Brazil

• Document type: Journal article
• Source: CONTRIBUTIONS TO DISCRETE MATHEMATICS; v. 6, n. 1, p. 154-159, 2011.
• Web of Science Citations: 0

Abstract

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. (AU)

• FAPESP's process: 04/14107-2 - Logical consequence and combinations of logics: fundaments and efficient applications
• Grantee: Walter Alexandre Carnielli
• Support Opportunities: Research Projects - Thematic Grants