I am reading the short paper due to Erdös and Bollobás "On a Ramsey-Turán type problem", where they obtain a lower bound for the number of edges on an $n$-graph without $K_4$ as a subgraph and small independence number (namely, $o(n)$).

The construction is geometric and uses that for given $k$, there is a sufficiently large non-negative integer $n$ such that the sphere $S^{k+1}=\{x\in R^{k+2}: \|x\|=1\}$ can be divided into $n$ sets 1) having equal measure and 2) small diameter.

This recall me Borsuk's conjecture, but not sure how to get this result, as here the number of parts is larger and we need that all the pieces have the same measure.

I assume that this is a folklore result, but I do not know where to find a reference for that.