Dense families of countable sets below $c$

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $\kappa$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $\kappa$, (2) $|\bar{a}\cap T|=\kappa$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $\kappa$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum ${\omega,\kappa}$.

💡 Research Summary

The paper establishes a consistency result concerning the structure of countable families below the continuum. Working in a model of ZFC where the Generalized Continuum Hypothesis holds, the authors perform a finite‑support iteration of c.c.c. forcing that adds λ many Cohen reals, where λ can be any cardinal as large as desired. This forcing raises the size of the continuum to λ while preserving all smaller cardinals, in particular any uncountable κ<λ.

In the resulting extension V