
Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
Posted:
Nov 4, 2013 6:00 PM


Working a bit further on I've hit another problem. The proof of the lemma about the sets P,P',Q and Q' also seems flawed. The definition of g is sufficient to give the desired result if P and P' are disjoint. E is presumably included to extend that to the case where they intersect, but I can't see how it does. g still only addresses disjoint subsets of P u P' u E. I think that can be worked round by altering the definition of g. Instead of listing the ways in which each set of size 2r can be split into disjoint sets of size r with the same image under f, it should give the ways in which it can yield two not necessarily disjoint sets of size r with the same image under f. The sets B' and B derived from this new g by Ramsey's theorem will still have any properties required for the rest of the proof, as the old g will be constant on any set that the new g is constant on.
And now I think I may see a way around the original problem.
Suppose D(x,y,i) but fx = fy. By the P/Q lemma, fx=fz for any other z such that D(x,z,i) [P = P' = x, Q = y, Q' = z]
There's a possible problem if x_i < y_i but z_i < x_i. The lemma as stated won't apply but the proof of it will extend to cover this case.
Now looking back at a rho not in L. By the definition
there exist distinct x,y such that D(x,y,i) and f(x) = f(y)
By the P/Q lemma, for all x,y such that D(x,y,i), f(x) = f(y)
which is what we needed.
So it seems the problem step can be justified using a lemma from later in the paper and, unless I've gone astray, that lemma can be proved if we modify g. That's all I have time for tonight.  David Hartley

