Banach Spaces and Ramsey Theory — Problem 2

Definitions and mathematical statement

The supplied record names $K(\delta)$ the Elton unconditionality constant, but does not supply its definition. With this source notation, the question is whether \[ \exists M\in{\mathbb{R}}\quad\left(M\geq 1\ \text{and}\ \forall\delta\in{\mathbb{R}}\; (0<\delta\leq 1\Longrightarrow K(\delta)\leq M)\right). \] Thus the requested bound must be independent of $\delta$, including arbitrarily small positive $\delta$ and the endpoint $\delta=1$. The definition of $K(\delta)$, including its underlying class of Banach spaces or sequences, scalar field, normalization, and defining quantifiers, is absent from the supplied record. Consequently this is a transcription of the source question, and a self-contained mathematical formulation remains unresolved pending that definition.

Short English statement

Is there a single finite constant $M\geq 1$ bounding the Elton unconditionality constants $K(\delta)$ for every $0<\delta\leq 1$? The supplied record labels this question open, but does not define $K(\delta)$.

Sources

• Local UnsolvedMath registry, permanent ID $800739$, source code catalogue entry, title Banach Spaces and Ramsey Theory --- Problem 2. The supplied status is open; this status has not been independently verified.
• Dodos and Lopez-Abad, Banach Spaces and Ramsey Theory: Some Open Problems (2010), Problem 2. Supplied source URL: https://arxiv.org/abs/1006.2668. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The problem text attributes the Elton unconditionality constant question to S. J. Dilworth, E. Odell, Th. Schlumprecht, and A. Zsák, with reference label [DOSZ]. The supplied record does not include the bibliographic entry corresponding to [DOSZ]. The registry lists Dodos and Lopez-Abad as proposers and gives the year 2010.