Uncountable equilateral sets in Banach spaces of the form \(C(K)\) (Q1650008)

A subset of a metric space is equilateral whenever any two points of it have the same distance. The existence of infinite equilateral sets in (infinite-dimensional) Banach spaces has been investigated for several years. The first example of a Banach space lacking such sets has been provided by \textit{P. Terenzi} [Boll. Unione Mat. Ital., VII. Ser., A 3, No. 1, 119--124 (1989; Zbl 0681.46025)], using a suitable renorming of $l_1$. Here, under a suitable combinatorial assumption and by means of forcing techniques, the author shows that the non-separable space $C(K)$ lacks uncountable equilateral sets for $K$ a version of the split interval obtained from a sequence of functions which behave in anti-Ramsey manner. That provides the first known example in the non-separable setting. Moreover, the author proves that, under extra set theoretic assumptions (Martin's axiom and the negation of the continuum hypothesis), for any non-metrizable compact Hausdorff space $K$, the space $C(K)$ contains some uncountable equilateral set.