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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

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 David Hartley Posts: 463 Registered: 12/13/04
Re: Another not-completely-insignificant gap in the Rado paper
Posted: Nov 7, 2013 4:35 PM

In message <mvZqxm9Vg\$eSFw0i@212648.invalid>, David Hartley
<me9@privacy.net> writes
>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.

This suggests a possible alternative proof of part (b).

Suppose {x_0,..,x_(r-1)}_< and {x'_0,...,x'_(r-1)}_< are subsets of B
which differ at an index which is in L. We need to show f(X) =/= f(X').

Lemma 1. If {x_0,..,x_(r-1)}_< and {x'_0,...,x'_(r-1)}_< c B; rho in L
and
x'_rho =/= x_i for any i

then f({x_0,..,x_(r-1)}) =/= f({x'_0,..,x'_(r-1)})

Proof. Put X = {x_0,..,x_(r-1)} and X' = {x'_0,..,x'_(r-1)}.

x'_rho is in B, so equals b_2p for some p. Let

x"_rho = b_(2r+1), x"_i = x'_i for all other i

and let X" = {x"_0,..,x"_(r-1)}

Now (X,X') = (X,X") so if f(X) = f(X') then f(X) = f(X"). But since rho
is in L, f(X') =/= f(X"). Hence f(X) =/= f(X')

Now there may not be index fitting the requirements of the lemma. In
that case let rho be the largest member of L where X and X' differ.
Wlog,

x_rho < x'_rho = x_sigma for some sigma > rho

Replace X and X' by X_0 and X_1 from [B(4)]^r such that (X,X') =
(X_0,X_1) and f(X) = f(X_0), f(X') = f(X_1)(as in Rado's proof). Let
X_0^sigma = b_4p. Form X_2 from X_0 by replacing X_0^sigma by b_(4p+2).
Since sigma is not in L, f(X_0) = f(X_2). X_1^rho = X_0^sigma is not in
X_2 so the lemma can be applied to X_2 and X_1, giving f(X_2) =/= f(x_1)
and so f(X_0) =/= f(X_1).

Hope there's no mistakes.

--
David Hartley

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
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11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/5/13 David Hartley
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11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
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11/7/13 David Hartley
11/7/13 Paul
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11/8/13 Paul
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11/7/13 Paul
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11/10/13 Paul
11/10/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
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11/4/13 Paul
11/4/13 Peter Percival