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From: Osher Doctorow on 25 Jun 2010 01:24
From Osher Doctorow

The Null Set (written N here) is defined as:

1) Null set = N = (definition) the set with 0 elements.

A Singleton (s) is defined as:

2) A singleton = s = (definition) a set with 1 element.

Notice that if we ERASE (symbol E as an "operator") the element of a
singleton, then we get the Null set. Symbolically:

3) E(s) = N

The USA and Canadian researchers Su Gao (U. North Texas USA), rnold W.
Miller (U. Madison Wisconsin USA) and William A. R. Weiss (U. Toronto
Canada) in 2006 came up with some remarkable differences between Null
Sets, Singletons, sets of 2 or more points, and relationships with
differences x - y of elements or quantities. Their paper is:

4) "Steinhaus sets and Jackson sets," arXiv: math/0603235 v1 [math.LO]
9 Mar 2006.

I will let readers look at their paper before commenting much further,
except to mention a few things. The Null Set N and a Singleton s are
"opposites" in some remarkable ways, such as:

5) The Null set is Jackson and a Singleton is never Jackson. Here a
set X is "Jackson set in R^n if there does not exist any Steinhaus Set
for X in R^n, where set S in R^n is Steinhaus for X where X is a
subset of R^n if |YS| (the cardinality of the intersection of Y and S)
= 1 for Y any isometric copy of X in R^n. 2 and 3 point sets in R^n
are Jackson under fairly general conditions. for n > = 2.

It turns out that the set of differences of elements or quantities
under specified conditions plays a key role in these and similar
relationships, including:

6) D* = D*(a1, ..., an) = {x - y| x, y are elements of A(a1,...,an)}
where (a1,...,an) is a sequence of positive integers and A(a1,...,an)
= {0, a1, a1+a2, ..., a1 + a2 + ... + an} for n > = 2.

Steinhaus turns out to be related to the non-existence of differences
of above type mod M (for M a "period"), while Jackson can have
differences. In addition, things turn out to be related to the
CHROMATIC NUMBER of a graph (see Wikipedia's or Wolfram's "Chromatic
number" online).

Osher Doctorow




From: Osher Doctorow on 25 Jun 2010 01:37
From Osher Doctorow

The second author is Arnold W. Miller, not rnold W. Miller.

It turns out that being Steinhaus for a set F and the Chromatic Number
of a certain graph related to F being less than or equal to the
cardinality of F are equivalent. The graph whose Chromatic number is
being considered is G(R^n, D(F)) where D(F) = {d(p, q): p, q are
unequal elements of F where F is a finite subset of R^n and the graph
G(R^n, D(F)) has vertices of R^n, p, q which are connected by an edge
if d(p, q) is in D(F).

Osher Doctorow
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