  We show that there exists an absolute constant $c_0<1$ such that for all $n
\ge 2$, any set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$
pairwise orthogonal vectors. The result is sharp up to the value of the
constant $c_0$. The proof relies on the harmonic analysis on the sphere and
hypercontractivity in the Gaussian space.
