Non-existence of low rank Ulrich bundles on Veronese varieties

We show that Veronese varieties of dimension $n\ge 4$ do not carry any Ulrich bundles of rank $r\le 3$. In order to prove this, we prove that a Veronese embedding of a complete intersection of dimension $m\ge 4$, which if $m=4$ is either ${\mathbb{P}}^4$ or has degree $d\ge 2$ and is very general and not of type $(2),(2,2)$, does not carry any Ulrich bundles of rank $r\le 3$.