Solution
= Solution
Order the $n$ sample points. Each of the $m-1$ cut points can occupy one of at most $n+1$ gaps. Once those gaps are chosen the fixed $\beta_k$ determine every sign, so
$$
\boxed{s(H_{F_1},n)\le(n+1)^{m-1}.}
$$
= Solution
Order the $n$ sample points. Each of the $m-1$ cut points can occupy one of at most $n+1$ gaps. Once those gaps are chosen the fixed $\beta_k$ determine every sign, so
$$
\boxed{s(H_{F_1},n)\le(n+1)^{m-1}.}
$$