  We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a
preference list, i.e., an ordering of the houses. A house allocation is an
injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than
another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$
does not come before $\tau(i)$ in the preference list of $i$. A house
allocation is Pareto optimal if there is no strictly better house allocation.
  Let $s(\tau)$ be the image of $\tau$ (i.e., the set of houses sold in the
house allocation $\tau$). We are interested in the largest possible cardinality
$f(m)$ of the family of sets $s(\tau)$ for Pareto optimal mappings $\tau$ taken
over all sets of preference lists of $m$ buyers. We improve the earlier upper
bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection
between this problem and some problems in extremal set theory.
