Monocolored topological complete graphs in colorings of uncountable complete graphs

P. Komjáth; S. Shelah

doi:10.1007/s10474-020-01125-3

Monocolored topological complete graphs in colorings of uncountable complete graphs

P. Komjáth^*
, S. Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

If κ> ℵ then κ→(κ,TopKκ)2, i.e., every graph on κ vertices contains either an independent set of κ vertices, or a topological K_κ, iff κ is regular and there is no κ-Suslin tree. Concerning the statement ω2→(TopKω2)ω2, i.e., in every coloring of the edges of Kω2 with countably many colors, there is a monochromatic topological Kω2, both the statement and its negation are consistent with the Generalized Continuum Hypothesis.

Original language	English
Pages (from-to)	71-84
Number of pages	14
Journal	Acta Mathematica Hungarica
Volume	163
Issue number	1
DOIs	https://doi.org/10.1007/s10474-020-01125-3
State	Published - Feb 2021

Bibliographical note

Publisher Copyright:
© 2021, Akadémiai Kiadó, Budapest, Hungary.

Keywords

Suslin tree
forcing
partition relation

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10.1007/s10474-020-01125-3

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abstract = "If κ> ℵ then κ→(κ,TopKκ)2, i.e., every graph on κ vertices contains either an independent set of κ vertices, or a topological Kκ, iff κ is regular and there is no κ-Suslin tree. Concerning the statement ω2→(TopKω2)ω2, i.e., in every coloring of the edges of Kω2 with countably many colors, there is a monochromatic topological Kω2, both the statement and its negation are consistent with the Generalized Continuum Hypothesis.",

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AB - If κ> ℵ then κ→(κ,TopKκ)2, i.e., every graph on κ vertices contains either an independent set of κ vertices, or a topological Kκ, iff κ is regular and there is no κ-Suslin tree. Concerning the statement ω2→(TopKω2)ω2, i.e., in every coloring of the edges of Kω2 with countably many colors, there is a monochromatic topological Kω2, both the statement and its negation are consistent with the Generalized Continuum Hypothesis.

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KW - partition relation

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Monocolored topological complete graphs in colorings of uncountable complete graphs

Abstract

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Keywords

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