Rado's Conjecture and the random algebra

Abstract

Rado's Conjecture (RC) is a compactness principle for a certain class of partial orders, namely trees $T$ of height $\omega_1$ without cofinal branches, postulating that a partial order $P$ from this class can be decomposed into at most countably many antichains if and only if all its suborders of size $\omega_1$ can be decomposed into at most countably many antichains. Rado's Conjecture is thus an uncountable version of Mirsky's theorem asserting that for every natural number $n$, every infinite partial order $P$ can be decomposed into at most $n$ many antichains if and only if all its finite suborders can be decomposed into at most $n$ many antichains. Todorcevic showed that RC is consistent modulo a strongly compact cardinal. RC implies $2^\omega \le \omega_2$, and has powerful consequences such as the Singular Cardinal Hypothesis, the failure of $\square(\kappa)$ for every regular $\kappa \ge \omega_2$ (and hence in particular the Projective Determinacy), and the Strong Chang Conjecture. It is also known that it is incompatible with Martin Axiom. We show that RC is consistent with $2^\omega = \omega_2$ and the cardinal invariants in Cichon diagram corresponding to forcing with the random algebra, i.e., $\mathfrak{d} = \omega_1$, $\mathrm{cov}(\mathcal{N}) = \omega_2$, $\mathrm{non}(\mathcal{N}) = \omega_1$. This provides a new pattern of cardinal invariants known to be consistent with RC. To prove the theorem, we first observe that random algebras do not specialize non-special trees of height $\omega_1$ without cofinal branches. Then we use the random algebra $\mathcal{B}_\kappa$ for a strongly compact $\kappa$ to define a new version of Mitchell forcing which yields the required result.