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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 + ε)separated set in the unit sphere for any ε > 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.

Abstract. It is well known that if K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space C(K) of continuous functions on K has complemented copies of c 0 , i.e.,We address the question if this could be the only type of decompositions of C(K) ∼ c 0 into infinite-dimensional summands for K infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

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Abstract. We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ∞/c 0 is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.We also prove that it is consistent that there are universal Banach spaces of density c, but ℓ∞/c 0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into ℓ∞/c 0 .

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We construct a continuous image of a Radon-Nikodým compact space which is not Radon-Nikodým compact, solving the problem posed in the 80ties by Isaac Namioka.

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