Solution
= Solution
The functions $x\mapsto x^T\beta$ form a real vector space of dimension $p$. Hence the <VC dimension of a vector space> gives
$$
\operatorname{VC}(\mathcal H_{\mathcal F_1})\leq p.
$$
Applying the <Sauer-Shelah lemma> and then the <Sauer-Shelah growth bound>,
$$
s(\mathcal H_{\mathcal F_1},n)
\leq\sum_{j=0}^p\binom nj
\leq\boxed{(n+1)^p}.
$$
This is the <growth bound for homogeneous linear classifiers>.