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Stationary common-neighborhood properties and partition hypotheses
Abstract
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)2ω and PH1(ω2). If κ is Tκ+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 check H2(ω3,Ad)≠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.
Related
BibTeX
@misc{li2026stationary,
title = {{Stationary common-neighborhood properties and partition hypotheses}},
author = {Xiang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25092},
eprint = {2609.25092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25092},
}
RIS
TY - UNPB
TI - Stationary common-neighborhood properties and partition hypotheses
AU - Xiang Li
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.25092
UR - https://arxiv.org/abs/2609.25092
AB - We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact κ, Coll(ω_1,<κ) forces ω_2→_hc,<5(ω_2)^2_ω and PH_1(ω_2). If κ is T^κ^+_ω_1-Ramsey, the same collapse forces that every countable coloring of [ω_2]^2 has a stationary set X⊆ω_2 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 ω_3 and PH_1(ω_3), in which check H^2(ω_3,A_d)≠0 for every nontrivial abelian group A. This separates PH_1(ω_3) from PH_2(ω_3), with the exact consistency strength of one weakly compact cardinal. We also show that PH_1(ω_2×ω_5) is equiconsistent with two weakly compact cardinals.
ER -