  We study the algorithmic properties of first-order monomodal logics of frames
$\langle \mathbb{N}, \leq \rangle$, $\langle \mathbb{N}, < \rangle$, $\langle
\mathbb{Q}, \leq \rangle$, $\langle \mathbb{Q}, < \rangle$, $\langle
\mathbb{R}, \leq \rangle$, $\langle \mathbb{R}, < \rangle$, as well as some
related logics, in languages with restrictions on the number of individual
variables as well as the number and arity of predicate letters. We show that
the logics of frames based on $\mathbb{N}$ are $\Pi^1_1$-hard -- thus, not
recursively enumerable -- in languages with two individual variables, one
monadic predicate letter and one proposition letter. We also show that the
logics of frames based on $\mathbb{Q}$ and $\mathbb{R}$ are $\Sigma^0_1$-hard
in languages with the same restrictions. Similar results are obtained for a
number of related logics.
