Application of Borsuk-Ulam Theorem on graph

Question

On a finite Borsuk Graph, we take a finite subset $S$ of $S^k$,$k-$dimensional sphere, such that with every point $x\in S^k$ having $y\in S$ and with a property $d(x,y)<\varepsilon= 1-\frac{\delta}{2}$ to form an induced sub-graph $B_{\delta}^{k}[S]$.How is such set $S$ exist?