Two-cardinal Kurepa Hypotheses
Fanxin Wu
TL;DR
We analyze the two-cardinal Kurepa Hypotheses $\mathsf{KH}(\kappa,\lambda)$, establishing a key ZFC constraint $\lnot\mathsf{KH}(\kappa,\lambda)\land\mathsf{KH}(\kappa,\mu)\rightarrow\mathsf{KH}(\lambda^+,\mu)$ and mapping the consistency landscape for combinations of $\mathsf{KH}(\aleph_1,\aleph_1)$, $\mathsf{KH}(\aleph_1,\aleph_2)$, and $\mathsf{KH}(\aleph_2,\aleph_2)$. The paper develops forcing tools, notably $\mathbb{Q}(\kappa^+,\lambda,\mu)$, to create KH and a two-cardinal Silver-branch lemma to destroy KH, and situates these in a broader framework including CH-type consequences and scales for singular cardinals. It shows that all non-obvious KH combinations are consistent relative to large cardinals except for a single ruled-out pattern, and extends Cummings’ Aronszajn–Kurepa results to successors of singular cardinals by combining Mitchell forcing with Prikry-type singularization. The results provide a near-complete map of the consistency landscape for KH across small cardinals and connect KH phenomena to Chang’s Conjecture and pcf-scale behavior, highlighting a robust forcing approach for two-cardinal tree principles.
Abstract
We consider the two-cardinal Kurepa Hypothesis $\mathsf{KH}(κ,λ)$. We observe that if $κ\leqλ<μ$ are infinite cardinals then $\lnot\mathsf{KH}(κ,λ)\land\mathsf{KH}(κ,μ)\rightarrow\mathsf{KH}(λ^+,μ)$, and show that in some sense this is the only $\mathsf{ZFC}$ constraint. The case of singular $λ$ and its relation to Chang's Conjecture and scales is discussed. We also extend an independence result about Kurepa and Aronszajn trees due to Cummings to the case of successors of singular cardinal.