Xiang Li

From an ω₁-Ramsey cardinal we force that every countable coloring of [ω₂]² has 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 resulting monochromatic graph has diameter at most two after any nonstationary deletion, answering a question of Hrušák–Shelah–Zhang. From a Ramsey cardinal, we force the partition hypothesis PH₁(ω₂), improving the huge-cardinal upper bound recorded by Bannister–Bergfalk–Moore–Todorcevic. The proofs share a local normal-ideal construction. The stationary argument uses countably closed models, whereas the Ramsey argument uses pointwise fusion and external countable completeness of the ultrafilter. Finally, for nonempty directed quasi-orders P ≤_T Q, every n < ω, and every cardinal λ, we prove that PH_n(Q, λ) implies PH_n(P, λ), answering the Tukey-transfer question of the same authors.