Mathematical Sciences Department Seminar
Chris Wells, Auburn University
Thursday, January 23rd
3-4pm
Stratton 202
Title: Mostly orthogonal vectors
Abstract: What is the maximum number of mutually orthogonal vectors that can coexist in $\mathbb{R}^n$?
Well, $n$, of course!
What if we weaken the condition: instead of requiring that all vectors be orthogonal, let's require only that among any three of the vectors, some pair are orthogonal.
How many vectors can we have now?
Observe that the union of any pair of orthogonal bases of $\mathbb{R}^n$ satisfies this condition, so we can have at least $2n$ many vectors.
Can we do any better?
This question of Erd\H{o}s was originally answered by Rosenfeld via a pretty tricky algebraic argument.
In this lecture, we will work through a different proof using only the Cauchy--Schwarz inequality and some basic facts from linear algebra!
Afterward, we will see how these same basic ideas can be used to tackle other related problems.