  For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph
with vertex-set consisting of all the $k$-element subsets of
$\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they
are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$
and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A
\subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$,
then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at
least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k
\choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the
behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple
of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum
degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has
$s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the
union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the
above result is that if $\mathcal{F}$ is a family of $k$-element subsets of
$\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there
exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least
$$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other
sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and
Kupavskii, using different methods, have recently proven closely related
results under the hypothesis that $n$ is at least quadratic in $k$.

Combinatorics