  In 2011, Meunier conjectured that for positive integers $n,k,r,s$ with $
k\geq 2$, $r\geq 2$, and $n\geq \max (\{r,s\})k$, the chromatic number of $s$
-stable $r$-uniform Kneser hypergraphs is equal to $\left\lceil \frac{n-\max
(\{r,s\})(k-1)}{r-1}\right\rceil $. It is a strengthened version of the
conjecture proposed by Ziegler (2002), and Alon, Drewnowski and \L uczak
(2009). The problem about the chromatic number of almost $s$-stable $r$
-uniform Kneser hypergraphs has also been introduced by Meunier (2011).
  For the $r=2$ case of the Meunier conjecture, Jonsson (2012) provided a
purely combinatorial proof to confirm the conjecture for $s\geq 4$ and $n$
sufficiently large, and by Chen (2015) for even $s$ and any $n$. The case $
s=3$ is completely open, even the chromatic number of the usual almost $s$
-stable Kneser graphs.
  In this paper, we obtain a topological lower bound for the chromatic number
of almost $s$-stable $r$-uniform Kneser hypergraphs via a different approach.
For the case $r=2$, we conclude that the chromatic number of almost $s$-stable
Kneser graphs is equal to $n-s(k-1)$ for all $s\geq 2$. Set $t=n-s(k-1)$. We
show that any proper coloring of an almost $s$-stable Kneser graph must contain
a completely multicolored complete bipartite subgraph $K_{\left\lceil
\frac{t}{2}\right\rceil \left\lfloor \frac{t}{2} \right\rfloor }$. It follows
that the local chromatic number of almost $s$ -stable Kneser graphs is at least
$\left\lceil \frac{t}{2}\right\rceil +1$. It is a strengthened result of
Simonyi and Tardos (2007), and Meunier's (2014) lower bound for almost
$s$-stable Kneser graphs.
