Orthogonal Pairs in Maps from the Sphere to the Circle

Abstract

We prove that, for any $f:S^2\to S^1$ and any $\epsilon >0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $\pi /2+\epsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

arXiv comment10 pages, 1 figure