On Monday, November 4, 2013 2:26:37 AM UTC, David Hartley wrote: > In message <email@example.com>, Paul > > <firstname.lastname@example.org> writes > > >We are given that rho_0 does not belong to L. However, L is defined by > > >a "for all" statement. So, for rho_0, the for-all statement is false > > >and we can find some yi and yi' to make f(z0, ..., z_r-1) = f(y0,..., > > >y_r-1) true. > > > > > >But the author is stating something much stronger -- that we can deduce > > >the equality for an arbitrary yi and yi'. > > > > I've never met Ramsey theory before, so I may also be missing something. > > But as far as I can see you are quite correct: this step is a > > non-sequitur. > > > > The solution is presumably to redefine L such that the deduction is > > valid. Ah! I think this ties in with something you've written in the > > other thread (which I haven't read carefully). You and I are > > interpreting his definition of L in the same way, and I can't see any > > other way of reading it. But he presumably intended to define L as the > > set of numbers rho < r which have the property > > > > there exist two ordered sets of r numbers differ only at the position > > indexed by rho which do not have the same image under f. > > > > (I'm not writing it in his style, what terrible notation.) > > > > More straightforwardly, L is the set of rho < r which do *not* have the > > property > > > > whenever two ordered sets of r numbers differ only at the position > > indexed by rho, then they have the same image under f. > > > > So for rho_0 not in L the step in question is valid. > > > > -- > > David Hartley
Thanks very much for this correction. I will use this corrected version of L and attempt to understand the rest of the paper. I agree that this corrected L gets past the blockage I initially complained about. I look forward to reading the rest of the paper and I hope that correcting L in this way doesn't cause problems further on in the proof.
Assuming L is wrong in this way, it's difficult to understand how the referees missed it. Perhaps they thought "Well, Rado's a great combinatorist so we don't really need to check it too carefully."
Very easy to understand how Rado could make the mistake. It's very easy to say something that's not what you mean. And once you have a psychological picture of something, it's very easy to just replace your own text by your own psychological picture. In other words, it's very hard to proofread your own work. That's why we need referees, and if they missed such a basic point, it's hard to accept that they were acting professionally.
Perhaps one solution is for the referee of an elementary paper to be a less-informed reader. Rado's paper is clearly intended as something a maths undergraduate should be able to handle. Therefore, surely maths undergraduates should be part of the proof-reading audience?