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Universal graphs without large bipartite subgraphs

Published online by Cambridge University Press:  26 February 2010

Péter Komjáth  and
János Pach
Péter Komjáth
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, Budapest, V. Reáltanoda u. 13–15, Hungary.
János Pach
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, Budapest, V. Reáltanoda u. 13–15, Hungary.
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Abstract

Let 1 ≤ α ≤ β ≤ γ be cardinals, and let denote the class of all graphs on γ vertices having no subgraph isomorphic to Kα,β. A graph is called universal if every can be embedded into Go as a subgraph. We prove that, if α < ω ≤ γ and the General Continuum Hypothesis is assumed, then has a universal element, if, and only if, (i) γ > ω or (ii) γ = ω, α = 1 and β ≤ 3. Using the Axiom of Constructibility, we also show that there does not exist a universal graph in .

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Type
Research Article
Information
Mathematika , Volume 31 , Issue 2 , December 1984 , pp. 282 - 290
Copyright
Copyright © University College London 1984

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