
Re: Another notcompletelyinsignificant gap in the Rado paper
Posted:
Nov 7, 2013 9:05 AM


In message <b829deebb3314460a7235d76a8ac54d0@googlegroups.com>, Paul <pepstein5@gmail.com> writes >So far, even when I appeal to large xi terms, I don't see enough space >between the elements to be sure of obtaining the relationships of the >form [X0, X1] = [X1, X2] etc. > I've only just started to look at the rest of the proof, but here's my first thoughts.
At each successive stage we're given a larger set to draw the elements of the next X_i from. B(r^(s1)) has (r1) elements between each member of B(r^s). That should be enough. (There could be a problem if the least element of X1 is b_0, but that can be avoided by choosing X_1 carefully.)
>However, I think I see the issue. As written, I don't see where he >uses the fact that B is a proper subset of B'. > >Therefore, perhaps the definition of B(t) is an error? Perhaps the >element at index j in the sequence B(t) is intended to mean the term at >index j in the C sequence where C refers to the sequence: b0, b2, b4, >b6....
... but I think you may be right there. As specified B(t) is not a subset of B for odd t.
>There does seem to be some small problem either with the paper, or my >understanding of the paper, because I see no place in the paper where >he uses the fact that he has removed the odd index elements from B'. > >Perhaps he redefined the b_i elements so that the i index now refers to >their position in B rather than in B' but he definitely needs to tell >the reader that he is doing this.
The definition of pi assumes that X_sigma0^rho0 has an even index, so it seems he's still using indexing in B' but assuming the X_i are all within B.
This is where he uses the fact that B =/= B'. b_(2pi+1) is not in B allows you to change X_sigma0 to Z_sigma0 as it can't already be another element of X_sigma0. But it has the same orderrelationship as B_2pi with all other elements of B and so does not change the orderrelationship of X_sigma0 with other X_i. In particular (X_0,X_sigma0) = (X_0,Z_sigma0)
>I see that you need to remove the odd elements because I don't get >enough spacing, but the construction still doesn't seem coherent >because the [X0, X1] steps simply don't work if you follow the >definitions literally.
So it seems it all works if you just change B(t) to {b_2t, b_4t,...}
 David Hartley

