  The paper is concerned with the problem whether a nonseparable Banach space
must contain an uncountable set of vectors such that the distances between
every two distinct vectors of the set are the same. Such sets are called
equilateral. We show that Martin's axiom and the negation of the continuum
hypothesis imply that every nonseparable Banach space of the form $C(K)$ has an
uncountable equilateral set. We also show that one cannot obtain such a result
without an additional set-theoretic assumption since we construct an example of
nonseparable Banach space of the form $C(K)$ which has no uncountable
equilateral set (or equivalently no uncountable $(1+\varepsilon)$-separated set
in the unit sphere for any $\varepsilon>0$) making another consistent
combinatorial assumption.
  The compact $K$ is a version of the split interval obtained from a sequence
of functions which behave in an anti-Ramsey manner. It remains open if there is
an absolute example of a nonseparable Banach space of the form different than
$C(K)$ which has no uncountable equilateral set. It follows from the results of
S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in
which it has an uncountable equilateral set. It remains open if there are
consistent examples which have no uncountable equilateral sets in any
equivalent renorming but it follows from the results of S. Todorcevic that it
is consistent that every nonseparable Banach space has an equivalent renorming
in which it has an uncountable equilateral set.
