### Research Experience for Undergraduates (REU) Seminar

Title: Good Ole Euclidean Plane, Don't We Know All About It?

Speaker: **Alexander Soifer**, Rutgers University

Date: June 29, 2004 11:30 - 12:30 pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Define a Unit Distance Plane as a graph U2 on the set of all points of
the plane R2 as its vertex set, with two points adjacent iff they are
distance 1 apart. The chromatic number of U2 is called the chromatic
number of the plane. It makes sense to talk about a distance graph when
its set of vertices belongs to a metric space, and two points are
adjacent iff the distance between them belongs to a given set of
distances.

We will discuss the problem of finding the chromatic number of the plane,
including the recent Conditional Chromatic Number Theorem, obtained by
Saharon Shelah and the presenter, 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 will look at examples of distance
graphs on the real line R, the plane R2, and the n-dimensional Euclidean
space Rn, whose chromatic number depends upon the system of axioms we
choose for set theory.