TY - JOUR
T1 - More Ramsey theory for highly connected monochromatic subgraphs
AU - Hrušak, Michael
AU - Shelah, Saharon
AU - Zhang, Jing
N1 - Publisher Copyright:
© The Author(s), 2023.
PY - 2024/12/1
Y1 - 2024/12/1
N2 - An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey's theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušak, and Shelah (2021, Acta Mathematica Hungarica 163, 309-322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: • for every regular cardinal, •. Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181-197), we also show that • is consistent with. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.
AB - An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey's theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušak, and Shelah (2021, Acta Mathematica Hungarica 163, 309-322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: • for every regular cardinal, •. Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181-197), we also show that • is consistent with. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.
KW - forcing
KW - Highly connected graph
KW - partition relations
KW - saturated ideal
UR - http://www.scopus.com/inward/record.url?scp=85179456807&partnerID=8YFLogxK
U2 - 10.4153/S0008414X23000767
DO - 10.4153/S0008414X23000767
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AN - SCOPUS:85179456807
SN - 0008-414X
VL - 76
SP - 2136
EP - 2150
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
IS - 6
ER -