  Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $
such that $r \le n / 2$, its vertices are all the subsets of
$[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the
corresponding subsets are disjoint. The Erd\H{o}s--Ko--Rado theorem states that
the size of the largest independent set in this graph is $\binom{n-1}{r-1}$.
Now let us delete each edge of the graph $K(n,r)$ with some fixed probability
$p$ independently of each other. Quite surprisingly, the independence number of
such random subgraph $K_p(n,r)$ of the graph $ K(n,r) $ is, with high
probability, the same as the independence number of the initial graph. This
phenomenon is called the stability of the independence number. This paper
concerns the independence number of random subgraphs of the graph $G(n,r,1)$,
which vertices are the same as in the Kneser graph and there is an edge between
two vertices if they intersect in exactly one element. We prove the stability
result for constant $r$ and $p=\frac{1}{2}$. To do that, we show that any
sufficiently large independent set in $G(n,r,1)$ has some predefined structure,
and this is a result of independent interest.
