Contractibility of Vietoris-Rips Complexes of dense subsets in (Rⁿ, ₁) via hyperconvex embeddings

We consider the contractibility of Vietoris-Rips complexes of dense subsets of (Rⁿ, ₁) with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each n natural there is a rₙ>0 so that the Vietoris-Rips complex of (Zⁿ, ₁) at scale r is contractible for all r rₙ. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space X and its connection to the Vietoris-Rips complex of X. In this manner, we provide positive answers to the question above for the case n=2 and 3.