  Many reaction networks arising in applications are multistationary, that is,
they have the capacity for more than one steady state; while some networks
exhibit absolute concentration robustness (ACR), which means that some species
concentration is the same at all steady states. Both multistationarity and ACR
are significant in biological settings, but only recently has attention focused
on the possibility for these properties to coexist. Our main result states that
such coexistence in at-most-bimolecular networks (which encompass most networks
arising in biology) requires at least $3$ species, $5$ complexes, and $3$
reactions. We prove additional bounds on the number of reactions for general
networks based on the number of linear conservation laws. Finally, we prove
that, outside of a few exceptional cases, ACR is equivalent to
non-multistationarity for bimolecular networks that are small (more precisely,
one-dimensional or up to two species). Our proofs involve analyses of systems
of sparse polynomials, and we also use classical results from chemical reaction
network theory.
