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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Embedding graphs into colored graphs


Authors: A. Hajnal and P. Komjáth
Journal: Trans. Amer. Math. Soc. 307 (1988), 395-409
MSC: Primary 05C55; Secondary 03E35, 04A20, 05A17
Corrigendum: Trans. Amer. Math. Soc. 332 (1992), null.
MathSciNet review: 936824
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Abstract: If $ X$ is a graph, $ \kappa $ a cardinal, then there is a graph $ Y$ such that if the vertex set of $ Y$ is $ \kappa $-colored, then there exists a monocolored induced copy of $ X$; moreover, if $ X$ does not contain a complete graph on $ \alpha $ vertices, neither does $ Y$. This may not be true, if we exclude noncomplete graphs as subgraphs.

It is consistent that there exists a graph $ X$ such that for every graph $ Y$ there is a two-coloring of the edges of $ Y$ such that there is no monocolored induced copy of $ X$. Similarly, a triangle-free $ X$ may exist such that every $ Y$ must contain an infinite complete graph, assuming that coloring $ Y$'s edges with countably many colors a monocolored copy of $ X$ always exists.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0936824-0
PII: S 0002-9947(1988)0936824-0
Keywords: Infinite graphs, Ramsey theory, independence, forcing, hypergraphs, chromatic number
Article copyright: © Copyright 1988 American Mathematical Society