Xiang Li

We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact κ, Coll(ω₁, <κ) forces ω₂ →_hc,<5 (ω₂)²_ω and PH₁(ω₂). If κ is T^κ+_ω1-Ramsey, the same collapse forces that every countable coloring of [ω₂]² has a stationary set X ⊆ ω₂ and a color i such that every finite subset of X has stationarily many color-i common neighbors in X. From one weakly compact cardinal, we obtain a model of the <5-edge relation at ω₃ and PH₁(ω₃), in which Ȟ²(ω₃, A_d) ≠ 0 for every nontrivial abelian group A. This separates PH₁(ω₃) from PH₂(ω₃), with the exact consistency strength of one weakly compact cardinal. We also show that PH₁(ω₂ × ω₅) is equiconsistent with two weakly compact cardinals.

I will discuss connections between variants of Ramsey relations and the Čech cohomology of ordinals. From one weakly compact cardinal, we obtain a model where every countable coloring of pairs from ω₃ has a monochromatic highly connected subgraph with ω₃ vertices. The model satisfies the first partition hypothesis, with vanishing first and nonvanishing second cohomology for every nontrivial abelian coefficient group. This separates the first two partition hypotheses. At ω₂, the corresponding construction simultaneously answers questions of Hrušák–Shelah–Zhang and Bannister–Bergfalk–Moore–Todorčević.

I will explain how the first partition hypothesis uniformly bounds finite errors in coherent families, allowing compactness to glue local functions into a global one. The talk emphasizes combinatorial ideas and assumes no background in partition relations or cohomology.