Chromatic numbers of graphs — large gaps
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,λ+)-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,λ+)-chromatic graph of size λ+.
Mathematics Subject Classification (2010)03E35 05C15 05C63
Unable to display preview. Download preview PDF.
- N. G. de Bruijn and P. Erdos: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54; Indagationes Math. 13 (1951), 369–373.Google Scholar
- P. Erdos and A. Hajnal: On chromatic number of infinite graphs, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), 83–98. Academic Press, New York, 1968.Google Scholar
- P. Komjáth: Subgraph chromatic number, in: Set theory (Piscataway, NJ, 1999), vol. 58 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 99–106. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
- A. Rinot: Jensen’s diamond principle and its relatives, in: Set theory and its applications, volume 533 of Contemp. Math., 125–156. Amer. Math. Soc., Providence, RI, 2001.Google Scholar
- S. Shelah: On incompactness for chromatic number of graphs, Acta Mathematica Hungarica, accepted.Google Scholar
- S. Todorčević: Comparing the continuum with the first two uncountable cardinals, in: Logic and scientific methods (Florence, 1995), vol. 259 of Synthese Lib., 145–155. Kluwer Acad. Publ., Dordrecht, 1997.Google Scholar