Embedding graphs into colored graphs

Hajnal, A.; Komjáth, P.

doi:10.1090/S0002-9947-1988-0936824-0

Embedding graphs into colored graphs
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Trans. Amer. Math. Soc. 307 (1988), 395-409 Request permission

Corrigendum: Trans. Amer. Math. Soc. 332 (1992), 475.

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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