TY - JOUR
T1 - Axiom of choice and chromatic number
T2 - Examples on the plane
AU - Soifer, Alexander
AU - Shelah, Saharon
PY - 2004/2
Y1 - 2004/2
N2 - In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct a distance graph G2 on the plane R2, thus coming much closer to the setting of the chromatic number of the plane problem. The chromatic number of G2 is 4 in the Zermelo-Fraenkel-Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. 92 Ser. 2 (1970) 1).
AB - In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct a distance graph G2 on the plane R2, thus coming much closer to the setting of the chromatic number of the plane problem. The chromatic number of G2 is 4 in the Zermelo-Fraenkel-Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. 92 Ser. 2 (1970) 1).
KW - Axiom of choice
KW - Axiomatic set theory
KW - Chromatic number
KW - Erdös problems and related topics to discrete geometry
KW - Euclidean Ramsey theory
KW - Graph theory
UR - http://www.scopus.com/inward/record.url?scp=1942509707&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2004.01.001
DO - 10.1016/j.jcta.2004.01.001
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AN - SCOPUS:1942509707
SN - 0097-3165
VL - 105
SP - 359
EP - 364
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 2
ER -