Alexander Soifer

Axiom of choice and chromatic number of the plane

Define a graph U on the set of all points of the plane R as its vertex set, with two points adjacent iff they are distance 1 apart. The graph U ought to be called unit distance plane, and its chromatic number w is called chromatic number of the plane. Finite subgraphs of U are called finite unit distance plane graphs. In 1950 the 18-year old Edward Nelson posed the problem of finding w (see the problem’s history in [Soi1]). A number of relevant results were obtained under additional restrictions on monochromatic sets (see surveys in [CFG,KW,Soi2,Soi3]). Falconer, for example, showed [F] that w is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in the general case, leaving us with an embarrassingly wide range for w being 4, 5, 6 or 7. In their fundamental 1951 paper [EB], Erdos and de Bruijn have shown that the chromatic number of the plane is attained on some finite subgraph. This result has naturally channeled much of research in the direction of finite unit distance graphs. One limitation of the Erdos–de-Bruijn result, however, has remained a low key: they ARTICLE IN PRESS

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 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).

A six-coloring of the plane

Abstract A six-coloring of the euclidean plane is constructed such that the distance 1 is not realized by any color except one, which does not realize the distance 1 5 .

Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts

In August 1987 I attended an inspiring talk by Paul Halmos at Chapman College in Orange, California. It was entitled “Some Problems You Can Solve, and Some You Cannot.” This problem was an example of a problem “you cannot solve.”

Open Problems Session

During the workshop Ramsey Theory Yesterday, Today and Tomorrow at Rutgers University on May 27-29, 2009, I offered a Problem Posing Session. All 30 participants of the workshop attended the session, and almost everyone came to the board and posed favorite open problems. The session was scheduled for an hour and lasted twice as long. I asked for problem submissions in writing for this volume. Below you will find all submitted problems (which is far from all the problems orally presented at the workshop). In addition, see many more open problems in the surveys of this volume. The survey by Ronald L. Graham and Eric Tressler, for one, consists entirely of open problems.

Axiom of choice and chromatic number of R n

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 difference graphs on the real plane R2 whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space Rn, whose chromatic number is a positive integer in the Zermelo-Fraenkelchoice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory.

Another six-coloring of the plane

Abstract A six-coloring of the Euclidean plane is constructed such that the distance 1 is not realized by any color except one, which does not realize the distance √2 - 1.

The Hadwiger–Nelson Problem

Inspired by the Four-Color Conjecture, the Hadwiger–Nelson Problem became one of the famous open problems of mathematics in its own rights. It has withstood all assaults for 65 years, and attracted many mathematicians from many fields, including Paul Erdős and Ronald L. Graham. John F. Nash admired this problem and chose it for the present book. In this chapter we will discuss this problem, its history and generalizations, several of the many related open problems, and the state of the art results.

Countable Countably — Indecomposable Abelian Groups, n-Decomposable for Any Finite n

At the latest by 1956 (please see [1]) L. Kulikov had an example of an abelian group G decomposable into a direct sum of n non-zero summands for any positive integer n and indecomposable into a direct sum of א 0 non-zero summands. His group G was torsion and had the cardinality of continuum. Is there a smaller example? More precisely: is there a countable א 0 — indecomposable abelian group, n-decomposable for any finite n? Since every countable reduced torsion group decomposes into a direct sum of א 0 summands, the author had to look into “the nearest” class, the class of countable reduced groups of the torsion-free rank 1. In 1972 he found there 2 examples (Example in [2], Model III in [3]). And the question arose: find all countable reduced torsion abelian groups T such that there exists an א 0-indecomposable extension G of T by a torsion-free group of the torsion-free rank 1. (Automatically such G is n-decomposable for any finite n.)

The Story of The Book

This chapter conveys, apparently for the first time, the story of the creation of the legendary two-volume “Moderne Algebra” by Van der Waerden and the role of Emil Artin in creating this book.