Do I have the right definition of VC dimension?
A family of hypothesis functions on domain $\cal X$ is a subset of $\{0,1\}^{\cal X}$. A family $H$ shatters a set $S \subseteq \cal X$ if for every subset $T \subseteq S$ there exists a function $h \in H$ such that $h(s) = 1_{s \in T}$ for all $s \in S$, that is, $h(s) = 1$ if $s \in T$ and $h(s) = 0$ if $s \in S \setminus T$.

A singleton family $H$ cannot shatter any non-empty set $S$. Indeed, suppose that $H = \{h\}$ shatters $S \neq \emptyset$, and take any $s \in S$. Taking $T = \emptyset$, we see that $h(s) = 0$, whereas taking $T = S$, we see that $h(s) = 1$.