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 \([m]=\{1, 2, \ldots , m\}\) $[m]=\{1,2,\dots ,m\}$ equipped with symmetric difference metric d, specifically, \({\mathcal {F}}^m_n\) ${\mathcal{F}}_n^m$ , \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}\) ${\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+1}^m$ , \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}\) ${\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+2}^m$ , and \({\mathcal {F}}_{\preceq A}^m\) ${\mathcal{F}}_{⪯A}^m$ . Here \({\mathcal {F}}^m_n\) ${\mathcal{F}}_n^m$ is the collection of size n subsets of [m] and \({\mathcal {F}}_{\preceq A}^m\) ${\mathcal{F}}_{⪯A}^m$ is the collection of subsets \(\preceq A\) $⪯A$ where \(\preceq \) $⪯$ is a total order on the collections of subsets of [m] and \(A\subseteq [m]\) $A\subseteq [m]$ (see the definition of \(\preceq \) $⪯$ in Sect. 1). We prove that the Vietoris–Rips complexes \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}^m_n, 2)\) $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m,2)$ and \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}, 2)\) $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+1}^m,2)$ are either contractible or homotopy equivalent to a wedge sum of \(S^2\) $S^2$ ’s; also, the complexes \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}, 2)\) $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^m\cup {\mathcal{F}}_{n+2}^m,2)$ and \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_{\preceq A}^m, 2)\) $\mathcal{V}\mathcal{R}({\mathcal{F}}_{⪯A}^m,2)$ are either contractible or homotopy equivalent to a wedge sum of \(S^3\) $S^3$ ’s. We provide inductive formulae for these homotopy types extending the result of Barmak about the independence complexes of Kneser graphs KG \(_{2, k}\) ${}_{2,k}$ and the result of Adamaszek and Adams about Vietoris–Rips complexes of hypercube graphs with scale 2.