We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado’s conjecture, Fodor-type reflection, Δ-reflection, Stationary-sets reflection, Martin’s Maximum, and a generalized Chang’s conjecture. This is accomplished by showing that, under GCH-type assumptions, instances of incompactness for the chromatic number can be derived from square-like principles that are compatible with large amounts of compactness.
In addition, we prove that, in contrast to the chromatic number, the coloring number does not admit arbitrarily large incompactness gaps.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
J. E. Baumgartner: Generic graph construction, J. Symbolic Logic 49 (1984), 234–240.
A. M. Brodsky and A. Rinot: A microscopic approach to Souslin-tree constructions, Part I, Ann. Pure Appl. Logic, 168 (2017), 1949–2007.
A. M. Brodsky and A. Rinot: A microscopic approach to Souslin-tree constructions, Part II, 2018, in preparation.
A. M. Brodsky and A. Rinot: Distributive Aronszajn trees. Fund. Math., to appear 2019. http://www.assafrinot.com/paper/29
J. Cummings and M. Magidor: Martin’s maximum and weak square, Proc. Amer. Math. Soc. 139 (2011), 3339–3348.
J. Cummings: Iterated forcing and elementary embeddings, in: Handbook of set theory. Vols. 1, 2, 3, 775–883. Springer, Dordrecht, 2010.
N. G. de Bruijn and P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 and Indagationes Math., 13 (1951), 369–373.
P. Erdős and A. Hajnal: On chromatic number of infinite graphs, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), 83–98. Academic Press, New York, 1968.
L. Fontanella and Y. Hayut: Square and delta Reflection, Ann. Pure Appl. Logic 167 (2016), 663–683.
S. Fuchino, I. Juhász, L. Soukup, Z. Szentmiklóssy and T. Usuba: Fodor-type Reflection principle and Reflection of metrizability and meta-Lindelöfness, Topology Appl. 157 (2010), 1415–1429.
M. Foreman and R. Laver: Some downwards transfer properties for N2, Adv. in Math. 67 (1988), 230–238.
M. Foreman, M. Magidor and S. Shelah: Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1988), 1–47.
S. Fuchino, H. Sakai, L. Soukup and T. Usuba: More about the Fodor-type Reflection principle, preprint, 2012.
S. Fuchino, H. Sakai, V. T. Perez and T. Usuba: Rado’s Conjecture and the Fodor-type Reflection Principle, 2017, in preparation.
F. Galvin: Chromatic numbers of subgraphs, Period. Math. Hungar. 4 (1973), 117–119.
Y. Hayut and C. Lambie-Hanson: Simultaneous stationary Reflection and square sequences, J. Math. Log., 17 (2017), 1750010.
A. Hajnal and A. Máté: Set mappings, partitions, and chromatic numbers, Studies in Logic and the Foundations of Mathematics 80 (1975), 347–379.
P. Komjáth: The colouring number, Proc. London Math. Soc. (3) 54 (1987), 1–14.
P. Komjáth: Consistency results on infinite graphs, Israel J. Math. 61 (1988), 285–294.
R. Laver: Making the supercompactness of k indestructible under k-directed closed forcing, Israel J. Math. 29 (1978), 385–388.
C. Lambie-Hanson: Squares and covering matrices, Ann. Pure Appl. Logic 165 (2014), 673–694.
T. Miyamoto: On the consistency strength of the FRP for the second uncountable cardinal, RIMS Kôkyûroku 1686 (2010), 80–92.
M. Magidor and S. Shelah: When does almost free imply free? (For groups, transversals, etc.), J. Amer. Math. Soc. 7 (1994), 769–830.
A. Rinot: Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), 293–314.
A. Rinot: The Ostaszewski square, and homogeneous Souslin trees, Israel J. Math. 199 (2014), 975–1012.
A. Rinot: Chromatic numbers of graphs - large gaps, Combinatorica 35 (2015), 215–233.
A. Rinot: Hedetniemi’s conjecture for uncountable graphs, J. Eur. Math. Soc. (JEMS) 19 (2017), 285–298.
A. Rinot: Higher Souslin trees and the GCH, revisited, Adv. Math. 311 (2017), 510–531.
A. Rinot: Same graph, different universe, Arch. Math. Logic, 56 (2017), 783–796.
S. Shelah: Notes on partition calculus, in: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, 1257–1276. Colloq. Math. Soc. János Bolyai, Vol. 10. North-Holland, Amsterdam, 1975.
S. Shelah: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), 319–349.
S. Shelah: Incompactness for chromatic numbers of graphs, in: A tribute to Paul Erdős, 361–371, Cambridge Univ. Press, Cambridge, 1990.
S. Shelah: Re ecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), 25–53.
S. Shelah: On incompactness for chromatic number of graphs, Acta Mathematica Hungarica 139 (2013), 363–371.
L. Soukup: On c+-chromatic graphs with small bounded subgraphs, Period. Math. Hungar. 21 (1990), 1–7.
S. Todorčević: On a conjecture of R. Rado, J. London Math. Soc. (2) 27 (1983), 1–8.
S. Todorčević: A note on the proper forcing axiom, in: Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Math., 209–218. Amer. Math. Soc., Providence, RI, 1984.
S. Todorčević: Conjectures of Rado and Chang and cardinal arithmetic, in: Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), volume 411 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 385–398. Kluwer Acad. Publ., Dordrecht, 1993.
S. Unger: Compactness for the chromatic number at Nω1+1, unpublished note, 2015.
About this article
Cite this article
Lambie-Hanson, C., Rinot, A. Reflection on the Coloring and Chromatic Numbers. Combinatorica 39, 165–214 (2019). https://doi.org/10.1007/s00493-017-3741-6
Mathematics Subject Classification (2010)