Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Surprise at my failure to resolve an issue in an elementary paper by Rado
Replies: 44   Last Post: Nov 10, 2013 12:23 PM

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Another not-completely-insignificant gap in the Rado paper
Posted: Nov 7, 2013 4:34 PM
 Plain Text Reply

On Thursday, November 7, 2013 8:29:09 PM UTC, David Hartley wrote:
> In message <1e7387e2-16fc-487f-be17-b490ed25dd39@googlegroups.com>, Paul
>
> <pepstein5@gmail.com> writes
>

> >Thanks for your contributions. I don't see things quite the same way.
>
> >I think that, as before, the author meant exactly what he said. All
>
> >indexing is fine and beginning at b_0 is fine. You may need sufficient
>
> >largeness of the x terms where largeness is defined in relation to the
>
> >indexing of the b terms. I disagree with you that we have enough
>
> >B(r^(s-1)) elements between the members of B(r ^ s). We sometimes
>
> >(unless we argue further) want r instead of r-1.
>
>
>
> It is crucial that X_sigma0 and X_0 be sunsets of B. Then we know
>
> X_sigma0^rho0 has an even index, 2pi, and that b_(2pi+1) will not be in
>
> X_0 or X_sigma0. That's the whole reason for using B
>
>
>
>
>
> There's nothing in the existing specification to stop you choosing X_0
>
> and X_1 with all odd indices (unless r is even of course). That shows
>
> immediately that using the B(t) as stated can't work.
>
>
>
> Starting with b_0 is not fine. If the smallest member of X_1 is b_0 and
>
> the smallest of X_0 is larger then there is no way to construct X_2.
>
> Starting B(t) at b_2t is one way around that. Another would be to
>
> specify that rho_0 is the smallest index at which the x_i and x'_i
>
> differ.
>
>
>
>
>
>
>
> You are quite right about not having enough room to do the construction.
>
> All the elements of X_1 could lie between two successive elements of
>
> X_0. Then all of X_2 would have to be between the corresponding elements
>
> of X_1 and there may be only (r-1) available.
>
>
>

> >Whenever we see a problem in the construction, we move around it by
>
> >using the Blass-Epstein argument to say that, because we're working
>
> >inside B', the problem (not being able to find enough B(r^(s-1)
>
> >elements) is inconsistent with the assumed inequality f(x...) =/=
>
> >f(x'...).
>
>
>
> There is a sense in which that applies here. If X_0 and X_1 are
>
> interleaved in that way, then we can jump straight to the last part of
>
> the argument. The point of the X_i sequence is to get us to the position
>
> where we can change the rho0-indexed element of X_sigma0 to the
>
> one-higher indexed b without disturbing the relative orderings with X_0.
>
> If X_0 and X_1 are disjoint then that can't happen anyway and we can
>
> just put sigma0 = 1 and finish.
>

Many thanks for your thoughts on this.

It would be interesting to compare and contrast with http://www.renyi.hu/~p_erdos/1950-01.pdf which proves the same thing. Currently, I haven't had a chance to digest either paper. I'm put off by maths where the readers have to fill in the details themselves. I enjoy it far more when I follow the author's reasoning.

That would not be a good approach if I were an academic mathematician. But I'm not so I'm taking the approach I enjoy best.

Can't wait to have attained mastery of this theorem of genius.

Paul Epstein

Date Subject Author
11/3/13 Paul
11/3/13 David Hartley
11/3/13 fom
11/3/13 fom
11/3/13 fom
11/4/13 fom
11/4/13 Paul
11/4/13 Paul
11/4/13 Peter Percival
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/6/13 Paul
11/6/13 Paul
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/8/13 Paul
11/8/13 David Hartley
11/7/13 Paul
11/7/13 fom
11/8/13 Paul
11/8/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 David Hartley
11/10/13 Paul
11/4/13 Paul
11/4/13 Peter Percival

© The Math Forum at NCTM 1994-2018. All Rights Reserved.