From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Partition of the plane
Date: Wed, 7 Mar 2001 15:20:49 -0600
Newsgroups: sci.math
Summary: Coloring points in the plane with 3 colors without monochromatic triangles

On 6 Mar 2001, Ahmed Fares wrote:

> Is there a partition of R^2 into infinite countable collection of
> disjoint sets such that no set contains the vertices of an 
> equilateral triangle ?

An easy partial answer: yes, if you assume the continuum hypothesis
2^aleph_0 = aleph_1. That's a special case of a general theorem proved
in an old (1970s) paper by Erdos, Hajnal, and Rothschild: if V is a
set of cardinality aleph_1, and if S is a collection of 3-point
subsets of V with the property that each 2-point subset of V is
contained in at most countably many members of S, then you can color
the points of V with countably many colors so that no member of S
contains 3 points of the same color. They do that by first
partitioning V into disjoint countable sets so that each member of S
has at least 2 of its 3 points in the same set of the partition.

-- 
People who don't have a sense of humor shouldn't try to be funny.