  The paper concerns the problem whether a nonseparable $\C(K)$ space must
contain a set of unit vectors whose cardinality equals to the density of
$\C(K)$ such that the distances between every two distinct vectors are always
greater than one. We prove that this is the case if the density is at most
continuum and we prove that for several classes of $\C(K)$ spaces (of arbitrary
density) it is even possible to find such a set which is $2$-equilateral; that
is, the distance between every two distinct vectors is exactly 2.
