Axiom of choice and chromatic number: examples on the plane
نویسندگان
چکیده
منابع مشابه
Axiom of choice and chromatic number: examples on the plane
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 choo...
متن کاملAxiom of choice and chromatic number of the plane
1950 the 18-year old Edward Nelson posed the problem of finding χ (see its history in [S]). A number of relevant results were obtained under additional restrictions on monochromatic sets. K. Falconer, for example, showed [F] that χ is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in general case, leaving us with embarrassingly...
متن کاملAxiom of choice and chromatic number of Rn
In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I 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 examples of a distance graph on the real line R and differe...
متن کاملThe fractional chromatic number of the plane
The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted χ(R). The problem was introduced in 1950, and shortly thereafter it was proved that 4 ≤ χ(R) ≤ 7. These bounds are both easy to prove, but after more than 60 years ...
متن کاملOn the Chromatic Number of Subsets of the Euclidean Plane
The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is between 4 and 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational pl...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2004
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2004.01.001