2-complexes with large homological systoles

We prove an upper bound on the size of homological systoles in $2$-dimensional simplicial complexes with $n$ vertices and $m$ faces. Applying the probabilistic method, we also prove the existence of complexes with essentially optimally large systoles, providing a nearly matching lower bound. The results contrast with the analogous story for the girth of a graph---in two dimensions, homological systoles can be polynomially large in $n$, over a wide range of $m$. Our main tool is an upper bound the number of triangulated surfaces with $v$ vertices and $f$ faces. Sharp asymptotics are known for the number of triangulations of the genus $g$ surface when $g$ is fixed, but this seems to be a new bound on the total number of combinatorial types of surfaces of all genus.