Lusin sets and uncountable Auerbach systems

Hajek, Kania and Russo showed that under CH there is an equivalent renorming of the Banach space c_0(omega_1) without uncountable Auerbach systems. It is not known whether extra set-theoretic assumptions can be dropped here. During the talk I will present (a slight modification of) the construction by Hajek, Kania and Russo to prove that the assumption of CH may be weakened to each of the following:

• the cofinality of the continuum equals omega_1;
• there is a strongly Lusin set of reals (i.e. a multi-dimensional version of the classical notion of a Lusin set).

This is a work in progress, so all remarks, comments and ideas are welcome.