Chromatic numbers of graphs — large gaps

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+.

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Rinot, A. Chromatic numbers of graphs — large gaps. Combinatorica 35, 215–233 (2015). https://doi.org/10.1007/s00493-014-3074-7

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Mathematics Subject Classification (2010)

  • 03E35
  • 05C15
  • 05C63