The first Cheeger constant of a simplex

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.