Suppose $L_m = \{x_1^nx_2^n\cdots x_m^n \mid n \in \mathbb{N}\} \subset \{x_1, \dotsc, x_m\}^*$. For what $m$ is $L_m$ context sensitive?
What do I currently know:
$L_m$ is regular only for $m = 1$.
$L_m$ is context free only for $m \in \{1, 2\}$
$L_m$ is recursively enumerable $\forall m \in \mathbb{N}$
Either $L_m$ is context sensitive $\forall n \in \mathbb{N}$ or $\exists m_0 \geq 3$ such that $L_m$ is context sensitive iff $m \leq m_0$.
$L_m$ is context-sensitive for all $m$. A Linear Bounded Automaton can, for example, mark the first unmarked $x_1$, then the first unmarked $x_2$ till the first unmarked $x_m$. Then it starts over. It accepts, if all of these runs mark exactly one of each of the types of symbols and after the last cycle no unmarked symbol is left. This clearly does not need more space than the input string.