  We study the fully dynamic maximum matching problem. In this problem, the
goal is to efficiently maintain an approximate maximum matching of a graph that
is subject to edge insertions and deletions. Our focus is particularly on
algorithms that maintain the edges of a $(1-\epsilon)$-approximate maximum
matching for an arbitrarily small constant $\epsilon > 0$. Until recently, the
fastest known algorithm for this problem required $\Theta(n)$ time per update
where $n$ is the number of vertices. This bound was slightly improved to
$n/(\log^* n)^{\Omega(1)}$ by Assadi, Behnezhad, Khanna, and Li [STOC'23] and
very recently to $n/2^{\Omega(\sqrt{\log n})}$ by Liu [ArXiv'24]. Whether this
can be improved to $n^{1-\Omega(1)}$ remains a major open problem.
  In this paper, we present a new algorithm that maintains a
$(1-\epsilon)$-approximate maximum matching. The update-time of our algorithm
is parametrized based on the density of a certain class of graphs that we call
Ordered Ruzsa-Szemer\'edi (ORS) graphs, a generalization of the well-known
Ruzsa-Szemer\'edi graphs. While determining the density of ORS (or RS) remains
a hard problem in combinatorics, we prove that if the existing constructions of
ORS graphs are optimal, then our algorithm runs in $n^{1/2+O(\epsilon)}$ time
for any fixed $\epsilon > 0$ which would be significantly faster than existing
near-linear in $n$ time algorithms.
