Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-13T12:28:32.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Forcing constructions for uncountably chromatic graphs

Published online by Cambridge University Press:  12 March 2014

Péter Komjáth  and
Saharon Shelah
Péter Komjáth
Affiliation:
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
Affiliation:
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
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 696 - 707
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Erdős, P., Problems and results on finite and infinite combinatorial analysis, Infinite and finite sets (P. Erdős sixtieth birthday colloquium, Keszthely, 1973; Hajnal, A.et al., editors), Colloquia Mathematica Societatis János Bolyai, vol. 10, part 1, North-Holland, Amsterdam, 1975, pp. 403–424.Google Scholar
[2]Erdős, P., Problems and results on finite and infinite combinatorial analysis. II, L'Enseignement Mathématique, vol. 27 (1981), pp. 163–176.Google Scholar
[3]Erdős, P., On the combinatorial problems I would most like to see solved, Combinatorica, vol. 1 (1981), pp. 24–42.CrossRefGoogle Scholar
[4]Erdős, P. and Hajnal, A., Unsolved problems in set theory, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 17–48.CrossRefGoogle Scholar
[5]Erdős, P. and Hajnal, A., Chromatic number of finite and infinite graphs and hypergraphs, Discrete Mathematics, vol. 53 (1985), pp. 281–285.CrossRefGoogle Scholar
[6]Folkman, J., Graphs with monochromatic complete subgraphs in every edge coloring, SIAM Journal of Applied Mathematics, vol. 18 (1970), pp. 115–124.CrossRefGoogle Scholar
[7]Hajnal, A., A negative partition relation, Proceedings of the National Academy of Sciences of the United States of America, vol. 68 (1971), pp. 142–144.CrossRefGoogle ScholarPubMed
[8]Hajnal, A. and Komjáth, P., What must and what need not be contained in every graph of uncountable chromatic number? Combinatorica, vol. 4 (1984), pp. 47–52.CrossRefGoogle Scholar
[9]Hajnal, A. and Máté, A., Set mappings, partitions, and chromatic numbers, Logic Colloquium '73, North-Holland, Amsterdam, 1975, pp. 347–379.Google Scholar
[10]Komjáth, P., A note on the Hajnal-Máté graphs, Studia Scientiarum Mathematicarum Hungarica, vol. 15 (1980), pp. 275–276.Google Scholar
[11]Komjáth, P., Mekler, A., and Pach, J., Universal graphs (to appear).Google Scholar
[12]Nešetřil, J. and Rödl, V., The Ramsey property for graphs with forbidden subgraphs, Journal of Combinatorial Theory, Series B, vol. 20 (1976), pp. 243–249.CrossRefGoogle Scholar
[13]Rödl, V., On the chromatic number of subgraphs of a graph, Proceedings of the American Mathematical Society, vol. 64 (1977), pp. 370–371.CrossRefGoogle Scholar
[14]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[15]Shelah, S., Can you take Solovay's inaccessible away? Israel Journal of Mathematics, vol. 48 (1984), pp. 1–47.CrossRefGoogle Scholar