On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

We examine the homotopy types of Vietoris-Rips complexes on certain finite metric spaces at scale $2$. We consider the collections of subsets of $[m]=\{1,2,\dots ,m\}$ equipped with symmetric difference metric $d$, specifically, ${\mathcal{F}}_n^m$, ${\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+1}^m$, ${\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+2}^m$, and ${\mathcal{F}}_{⪯A}^m$. Here ${\mathcal{F}}_n^m$ is the collection of size $n$ subsets of $[m]$ and ${\mathcal{F}}_{⪯A}^m$ is the collection of subsets $⪯A$ where $⪯$ is a total order on the collections of subsets of $[m]$ and $A\subseteq [m]$ (see the definition of $⪯$ in Section~\ref{Intro}). We prove that the Vietoris-Rips complexes $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m,2)$ and $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+1}^m,2)$ are either contractible or homotopy equivalent to a wedge sum of $S^2$’s; also, the complexes $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+2}^m,2)$ and $\mathcal{V}\mathcal{R}({\mathcal{F}}_{⪯A}^m,2)$ 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 $2$.