The following is stated in the Erdös-Bollobás Paper - On a Ramsey-Turán type Problem
If $n$ is a sufficiently large number, then $k+1$ dimensional sphere can be divided into $n$ sets, each of equal measure and diameter bounded by $\frac{\epsilon}{10\sqrt{k}}$
Is there any simple explicit construction that might help show this? I guess $n$ being really large ensure that the diameter can be made as small as possible, but I'm having trouble formalizing this logic. Particularly, I'm having trouble in splitting into ($n$) equal parts along with small diameter
Here's one simple approach. Suppose we want to chop up the hypersphere so that each piece has equal measure and fits inside an $h \times h \times \dots \times h$ hypercube.
First, find some hyperplanes whose $d^{\text{th}}$ coordinate is constant that chop up the hypersphere into slices of equal measure. This is possible for any number of hyperplanes, and as the number of slices goes to infinity, the thickness of each slice goes to $0$. So we can choose a number of slices large enough that the thickness is at most $h$.
Then, we can repeat this by dividing each slice with hyperplanes whose $(d-1)^{\text{th}}$ coordinate is constant, and which are all at most $h$ apart. The argument is the same, and the only tricky thing is that for each slice from the first step, there might be a different minimum number of times we want to divide it in the second step. But there's finitely many slices, so we can just take the greatest lower bound on the number of divisions necessary for each slice.
Then, repeat in each coordinate. At the end, we have finitely many pieces whose thickness in the $i^{\text{th}}$ dimension is at most $h$, for all $i$. Their diameters are at most $h\sqrt k$ as a result, and by taking $h = \frac{\epsilon}{10k}$, we get the division we wanted.