  The coboundary expansion generalizes the classical graph expansion to the
case of the general simplicial complexes, and allows the definition of the
higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial
complex $X$, and any $k\geq 0$. In this paper we investigate the value of
$h_1(\Delta^{[n]})$ - the first Cheeger constant of a simplex with $n$
vertices. It is known, due to the pioneering work of Meshulam and Wallach, that
\[\lceil n/3\rceil\geq h_1(\Delta^{[n]})\geq n/3, \textrm{ for all } n,\] and
that the equality $h_1(\Delta^{[n]})=n/3$ is achieved when $n$ is divisible by
$3$.
  Here we expand on these results. First, we show that \[h_1(\Delta^{[n]})=n/3,
\textrm{ whenever }n\textrm{ is not a power of }2.\] So the sharp equality
holds on a set whose density goes to $1$. Second, we show that
\[h_1(\Delta^{[n]})=n/3+O(1/n),\textrm{ when }n\textrm{ is a power of }2.\] In
other words, as $n$ goes to infinity, the value $h_1(\Delta^{[n]})-n/3$ is
either $0$ or goes to $0$ very rapidly.
  Our methods include recasting the original question in purely graph-theoretic
language, followed by a detailed investigation of a specific graph family, the
so-called {\it staircase graphs}. These are defined by associating a graph to
every partition, and appear to be especially suited to gain information about
the first Cheeger constant of a simplex.
