This is an announcement for the paper “Large separated sets of unit vectors in Banach spaces of continuous functions” by Marek Cúthhttps://arxiv.org/find/math/1/au:+Cuth_M/0/1/0/all/0/1, Benjamin Vejnarhttps://arxiv.org/find/math/1/au:+Vejnar_B/0/1/0/all/0/1, Ondřej Kurkahttps://arxiv.org/find/math/1/au:+Kurka_O/0/1/0/all/0/1.
Abstract: The paper is concerned with 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 $2$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.00478