  We show that every Banach space $X$ containing an isomorphic copy of $c_0$
has an infinite equilateral set and also that if $X$ has a bounded biorthogonal
system of size $\alpha$ then it can be renormed so as to admit an equilateral
set of equal size. If $K$ is any compact non metrizable space, then under a
certain combinatorial condition on $K$ the Banach space $C(K)$ has an
uncountable equilateral set.
