  Let $G$ be a connected graph on $n$ vertices. The Gallai number $Gal(G)$ of
$G$ is the size of the smallest set of vertices that meets every maximum path
in $G$. Gr\"unbaum constructed a graph $G$ with $Gal(G)=3$. Very recently,
Long, Milans, and Munaro, proved that $Gal(G)\leq 8n^{{3}/{4}}$. This was the
first sublinear upper bound on $Gal(G)$ in terms of $n$. We improve their bound
to $Gal(G)\leq 5 n^{{2}/{3}}$. We also tighten a more general result of Long et
al. For a multigraph $M$ on m edges, we prove that if the set $L(M,G)$ of
maximum $M$-subdivisions in $G$ is pairwise intersecting and $n\geq m^{6}$,
then $G$ has a set of vertices with size at most $5 n^{{2}/{3}}$ that meets
every $Q\in \mathcal{L}(M,G)$
