Combinatorica

, Volume 35, Issue 2, pp 215–233 | Cite as

Chromatic numbers of graphs — large gaps

Original Paper

Abstract

We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|.

The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+.

We also study (ℵ0+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0+)-chromatic graph of size λ+.

Mathematics Subject Classification (2010)

03E35 05C15 05C63 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.The Fields Institute for Research in Mathematical SciencesTorontoCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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