Stationary Common Neighbors and Partition Hypotheses

Collapsing a $T_{{\omega }_1}^{{\kappa }^{+}}$-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 Hrušák–Shelah–Zhang. Collapsing a weakly compact cardinal gives ${\mathrm{PH}}_1({\omega }_2)$ and, together with the known lower bound, determines its exact consistency strength. Both results use local seed ideals. We also prove that ${\mathrm{PH}}_n(Q,\lambda )\Rightarrow {\mathrm{PH}}_n(P,\lambda )$ whenever $P{\le }_{T}Q$ 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 $a\ge 2$ and $b\ge a+3$, a product lemma gives, from two weakly compact cardinals, the consistency of ${\mathrm{PH}}_1(P)$ for every nonempty directed quasi-order $P{\le }_T{\omega }_a\times {\omega }_b$.