Uncountable equilateral sets in Banach spaces

A set A in a Banach space X is said to be equilateral if there is a real r such that the distance between any two distinct elements of A is r.
We construct the first consistent example of a nonseparable Banach space without an uncountable equilateral set. The example is a Banach space of the form C(K) for K compact connected (If K is totally disconnected, then C(K) always has an equilateral set of the cardinality equal to the weight of K). K is a connected version of the split interval obtained from an "antiRamsey" collection of continuous functions f:([0,1]-{r})-->[0,1] which are obtained by an elementary forcing argument.

On the other hand we show that MA + the negation of CH implies that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. It remains open if there is an absolute example of a Banach space (of some other form than C(K)) without uncountable
equilateral sets.