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Let $A,B$ with $|A| = m$ and $|B| = n\ge m$ be two sets. We assume that everyelement $a\in A$ has a reference list over all elements from $B$. We call aninjective mapping $\tau$ from $A$ to $B$ a matching. A blocking coalition of$\tau$ is a subset $A'$ of $A$ such that there exists a matching $\tau'$ thatdiffers from $\tau$ only on elements of $A'$, and every element of $A'$improves in $\tau'$, compared to $\tau$ according to its preference list. Ifthere exists no blocking coalition, we call the matching $\tau$ an exchangestable matching (ESM). An element $b\in B$ is reachable if there exists anexchange stable matching using $b$. The set of all reachable elements isdenoted by $E^*$. We show \[|E^*| \leq \sum_{i = 1,\ldots,m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m).\] This isasymptotically tight. A set $E\subseteq B$ is reachable (respectively exactlyreachable) if there exists an exchange stable matching $\tau$ whose imagecontains $E$ as a subset (respectively equals $E$). We give bounds for thenumber of exactly reachable sets. We find that our results hold in the moregeneral setting of multi-matchings, when each element $a$ of $A$ is matchedwith $\ell_a$ elements of $B$ instead of just one. Further, we give complexityresults and algorithms for corresponding algorithmic questions. Finally, wecharacterize unavoidable elements, i.e., elements of $B$ that are used by allESM's. This yields efficient algorithms to determine all unavoidable elements.

Counting Houses of Pareto Optimal Matchings in the House Allocation Problem