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Forcing constructions for uncountably chromatic graphs

Published online by Cambridge University Press:  12 March 2014

Péter Komjáth
Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015 Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 Institute of Mathematics, The Hebrew University, Jerusalem, Israel
Saharon Shelah
Institute of Mathematics, Eötvös University, Budapest 1088, Hungary Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 Institute of Mathematics, The Hebrew University, Jerusalem, Israel


In this paper we solve some of Pál Erdős's favorite problems on uncountably chromatic graphs. Generalizing a finite graph theory result of Tutte, Erdős and R. Rado showed that for every infinite cardinal κ there exists a triangle-free, κ-chromatic graph of size κ. For κ = ℵ0, Erdős established the existence of ℵ0-chromatic graphs excluding even C4, C5,…, Cn, i.e. circuits up to a given length. For κ < ℵ0 the situation is different. As shown by Erdős and A. Hajnal, a graph is necessarily countably chromatic if it omits any finite bipartite graph. We can, however, exclude any finite list of nonbipartite graphs (this obviously reduces to excluding finitely many odd circuits). They posed an even stronger conjecture, namely, that similar examples must occur in every uncountably chromatic graph. To be specific, they conjectured that for every infinite κ, every κ-chromatic graph contains a κ-chromatic triangle-free subgraph. Here we show that this may not be true for κ = ℵ1 i.e. we exhibit a model where it is false. We must emphasize that the conjecture is probably false already in ZFC, but we have been unable to show this.

Research Article
Copyright © Association for Symbolic Logic 1988

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