  For $k,n\in \mathbb{N}$, the Kneser graph $K(n,k)$ is the graph with vertex
set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$.
Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian. Similarly as
for graphs with Hajnal's and Szemer\'edi's result about a minimum degree
condition for clique factors and the P\'osa-Seymour Conjecture together with
its solution for large graphs due to Koml\'os, S\'ark\"ozy, and Szemer\'edi,
the next step is to ask for clique factors and powers of Hamiltonian cycles in
Kneser graphs. For $k,\ell\in \mathbb{N}$, let $n(k,\ell)$ be the smallest
integer such that for $n\geq n(k,\ell)$, $K(n,k)$ contains the $\ell$-th power
of a Hamiltonian cycle. Katona conjectured that for all but finitely many
exceptions, $n(k,\ell)=(\ell+1)k+1$ holds. In particular, it would be
interesting to know whether $n(k,\ell)$ is linear in $k$ (for fixed $\ell$). So
far this is not known for $k\geq 2$. In this note, we take a first step towards
such a linear bound by proving that for $\ell\in \mathbb{N}$, $k\geq \ell$ and
$n\geq \ell ^3k$, all but at most $\ell-1$ vertices of $K(n,k)$ can be
partitioned into cliques of size $\ell$.
  Further, we use our methods to extend a short proof due to Chen and F\"uredi
that $K(n,k)$ is Hamiltonian for $n\geq 3k$ and $k \mid n$ to all $n\geq 4k$ if
$k\geq 4$.
