  We examine the homotopy types of Vietoris-Rips complexes on certain finite
metric spaces at scale $2$. We consider the collections of subsets of $[m]=\{1,
2, \ldots, m\}$ equipped with symmetric difference metric $d$, specifically,
$\mathcal{F}^m_n$, $\mathcal{F}_n^m\cup \mathcal{F}^m_{n+1}$,
$\mathcal{F}_n^m\cup \mathcal{F}^m_{n+2}$, and $\mathcal{F}_{\preceq A}^m$.
Here $\mathcal{F}^m_n$ is the collection of size $n$ subsets of $[m]$ and
$\mathcal{F}_{\preceq A}^m$ is the collection of subsets $\preceq A$ where
$\preceq$ is a total order on the collections of subsets of $[m]$ and
$A\subseteq [m]$ (see the definition of $\preceq$ in Section~\ref{Intro}). We
prove that the Vietoris-Rips complexes $\mathcal{VR}(\mathcal{F}^m_n, 2)$ and
$\mathcal{VR}(\mathcal{F}_n^m\cup \mathcal{F}^m_{n+1}, 2)$ are either
contractible or homotopy equivalent to a wedge sum of $S^2$'s; also, the
complexes $\mathcal{VR}(\mathcal{F}_n^m\cup \mathcal{F}^m_{n+2}, 2)$ and
$\mathcal{VR}(\mathcal{F}_{\preceq A}^m, 2)$ are either contractible or
homotopy equivalent to a wedge sum of $S^3$'s. We provide inductive formula for
these homotopy types extending the result of Barmak in \cite{Bar13} about the
independence complexes of Kneser graphs \text{KG}$_{2, k}$ and the result of
Adamaszek and Adams in \cite{AA22} about Vietoris-Rips complexes of hypercube
graphs with scale $2$.
