On antiramsey colorings of uncountable squares and geometry of nonseparable Banach spaces

A subset Z of a Banach space X is said to be r-equilateral (r-separated) if every two distinct elements of Z are in the distance exactly (at least) r from each other.
We will address the question of the existence of uncountable equilateral and (1 + e)-separated sets (e > 0) in the unit spheres of some nonseparable Banach spaces X induced by antiramsey colorings of pairs of countable ordinals.

The corollaries are that non(M) = \omega_1 implies
1) the existence of an equivalent renorming of the Hilbert space of density \omega_1 which does not admit any uncountable equilateral set
and
2) the existence of a nonseparable Hilbert generated Banach space containing an isomorphic copy of l_2 in each nonseparable subspace, whose unit sphere does not admit an uncountable equilateral set and does not admit an uncountable (1 + e)-separated set for any e > 0.

It turns out that in our approach additional set-theoretic axioms are inevitable, since under (MA + \neg CH) the geometry of the spaces under consideration is regular. The talk is based on a joint work with Piotr Koszmider:
https://arxiv.org/abs/2301.07413

This will be the last talk of this semester.