Let
$X$ be an
$n$element
set, where
$n$
is even. We refute a conjecture of J. Gordon and Y. Teplitskaya,
according to which, for every maximal intersecting family
$\mathcal{\mathcal{F}}$ of
$\frac{n}{2}$element
subsets of
$X$, one
can partition $X$
into
$\frac{n}{2}$
disjoint pairs in such a way that no matter how we pick one element from each of the
first
$\frac{n}{2}1$
pairs, the set formed by them can always be completed to a member of
$\mathcal{\mathcal{F}}$ by
adding an element of the last pair.
The above problem is related to classical questions in extremal set theory. For any
$t\ge 2$, we call a
family of sets
$\mathcal{\mathcal{F}}\subset {2}^{X}$
$t$separable if there is
a
$t$element subset
$T\subseteq X$ such that for every
ordered pair of elements
$\left(x,y\right)$
of
$T$, there
exists
$F\in \mathcal{\mathcal{F}}$ such
that
$F\cap \left\{x,y\right\}=\left\{x\right\}$. For
a fixed
$t$,
$2\le t\le 5$, and
$n\to \infty $,
we establish asymptotically tight estimates for the smallest integer
$s=s\left(n,t\right)$ such that
every family
$\mathcal{\mathcal{F}}$
with
$\left\mathcal{\mathcal{F}}\right\ge s$ is
$t$separable.
