Concentrated sets and $\gamma$-sets in the Miller model

TL;DR

This paper demonstrates that in the Miller model, there are no large concentrated sets or gamma-sets of reals of size 2, refuting a prior conjecture using combinatorial covering properties.

Contribution

It proves the non-existence of 2-sized concentrated sets and 2-sized b3-sets of reals in the Miller model, challenging previous assumptions.

Findings

No concentrated sets of size 2 in the Miller model

No b3-sets of size 2 in the Miller model

Refutes a conjecture of Bartoszynski and Halbeisen

Abstract

Using combinatorial covering properties, we show that there is no concentrated set of reals of size $\omega_2$ in the Miller model. The main result refutes a conjecture of Bartoszy\'{n}ski and Halbeisen. We also prove that there are no $\gamma$-set of reals of size $\omega_2$ in the Miller model.