  Bauwens and Zimand [BZ 2019] have shown that lossless expanders have an
interesting online matching property. The result appears in an implicit form in
[BZ 2019]. We present an explicit version of this property which is directly
amenable to typical applications, prove it in a self-contained manner that
clarifies the role of some parameters, and give two applications.
  A $(K, \epsilon)$ lossless expander is a bipartite graph such that any subset
$S$ of size at most $K$ of nodes on the left side of the bipartition has at
least $(1-\epsilon) D |S|$ neighbors, where $D$ is the left degree.The main
result is that any such graph, after a slight modification, admits
$(1-O(\epsilon)D, 1)$ online matching up to size $K$. This means that for any
sequence $S=(x_1, \ldots, x_K)$ of nodes on the left side of the bipartition,
one can assign in an online manner to each node $x_i$ in $S$ a set $A_i$
consisting of $(1-O(\epsilon))$ fraction of its neighbors so that the sets
$A_1, \ldots, A_K$ are pairwise disjoint. "Online manner" refers to the fact
that, for every $i$, the set of nodes assigned to $x_i$ only depends on the
nodes assigned to $x_1, \ldots, x_{i-1}$.
  The first application concerns storage schemes for representing a set $S$, so
that a membership query "Is $x \in S$?" can be answered probabilistically by
reading a single bit. All the previous one-probe storage schemes were for a
static set $S$. We show that a lossless expander can be used to construct a
one-probe storage scheme for dynamic sets, i.e., sets in which elements can be
inserted and deleted without affecting the representation of other elements.
The second application is about non-blocking networks.
