From - Mon Sep  1 17:09:57 1997
From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Newsgroups: sci.math
Subject: RAMSEY style question.
Date: 1 Sep 1997 05:11:04 GMT
Organization: Department of Mathematics and Statistics, 
University of Canterbury, Christchurch, NewZealand
Lines: 90

Some while ago there was a puzzle on some newsgroup, as follows:-

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Partition the integers 1 to 23 into three sets, such that for no set are
there three different numbers with two adding to the third.
===============

It doesn't seem to be *too* hard to find a solution by hand.
Solution (and further comments) below.

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There are three solutions...
*****************
   1   2   4   8   11   16   22
   3   5   6   7   19   21   23
   9   10   12   13   14   15   17   18   20
*****************
   1   2   4   8   11   17   22
   3   5   6   7   19   21   23
   9   10   12   13   14   15   16   18   20
*****************
   1   2   4   8   11   22
   3   5   6   7   19   21   23
   9   10   12   13   14   15   16   17   18   20
*****************

It's intriguing to note that most of the integers are fixed, with just
16 & 17 movable. (I doubt there's any real significance in this though.)

The article observed that 23 was the largest number for which this could
be done.  Very Ramsey-like.  More intriguing still, the solutions for
the similar maximal problems with 1 set and 2 sets are unique...

*****
1 2
*****

*********
1 2 4 8
3 5 6 7
*********

...with, most intriguing of all, the solutions for each case are
extensions of the solution for the previous case!

Does this feature continue, I wonder?

Surely this must be a standard Ramsey question?  What is known of this?
Can someone run off solutions (if feasible) for 4- and 5-part partitions?

Cheers,
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             Bill Taylor            W.Taylor@math.canterbury.ac.nz
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               If I have seen less far than other men,
               it is by standing in the footprints of giants.
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