Is the following TRUE or FALSE?
For every 17coloring of N (the naturals not including 0)
there exists x, y, z such that
x,y,z are distinct x,y,z that are same color such that
2x+3x6z = 0
It turns out that this is FALSE.We'll call a set b_{1},...,b_{n} REGULAR if
for every c, for every ccoloring of N, there exists
x_{1},....,x_{n} such thatx_{1},....,x_{n} are all the same color, andb_{1}x_{1} + ... + b_{n}x_{n} = 0
The following is known as (abridged) Rado's Theorem.
Rado proved it in 1933.
(b_{1},...,b_{n})
is regular iff some nonempty subset of the b_{i}'s
sum to 0.
For an exposition of the proof see Ramsey Theory
by Graham, Rothchild,and Spencer or see
my writeup
NOW here is a question to which the answer is known, and
I'll tell you the answer in my next post.
TRUE OR FALSE:
For every coloring of R (the reals) with a countable number of colors
there exists distinct x,y,z,wx,y,z,w same colorx+y=x+w.
When I asked this in seminar I got
5 thought it was TRUE

4 thought it was FALSE

5 thought it was a STUPID QUESTION.

Posted By GASARCH to Computational Complexity at 11/02/2007 02:26:00 PM