Stationary Common Neighbors and Partition Hypotheses

Abstract

Collapsing a $T^{\kappa^+}_{\omega_1}$-Ramsey cardinal $\kappa$ to $\omega_2$ gives, for every countable coloring of $[\omega_2]^2$, a stationary set $X$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. The color-$i$ graph on $X$ has diameter at most two after any nonstationary deletion, answering a question of Hru\v{s}\'ak--Shelah--Zhang. Collapsing a weakly compact cardinal gives $\operatorname{PH}_1(\omega_2)$ and, together with the known lower bound, determines its exact consistency strength. Both results use local seed ideals. We also prove that $\operatorname{PH}_n(Q,\lambda)\Rightarrow\operatorname{PH}_n(P,\lambda)$ whenever $P\leq_TQ$ are nonempty directed quasi-orders, for every $n<\omega$ and cardinal $\lambda$, answering the Tukey-transfer question of Bannister--Bergfalk--Moore--Todorcevic. Finally, for each pair of integers $a\geq2$ and $b\geq a+3$, a product lemma gives, from two weakly compact cardinals, the consistency of $\operatorname{PH}_1(P)$ for every nonempty directed quasi-order $P\leq_T\omega_a\times\omega_b$.