TalkRamThyEnLB - 1 RAMSEY THEORY AND ENORMOUS LOWER BOUNDS...

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1 RAMSEY THEORY AND ENORMOUS LOWER BOUNDS Abstract by Harvey M. Friedman Department of Mathematics Ohio State University [email protected] www.math.ohio-state.edu/~friedman/ April 5, 1997 Not everyone realizes that the classical Ramsey theorem's were stated and proved in order to solve a problem in mathematical logic. This can be seen by reading the title of the classical paper, F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286. The original logic problem was solved by Ramsey in that paper, and it was a decision problem. Although Ramsey's theorem is used in an essential way, this decision problem has a nondeterministic exponential time procedure, which is not an enormous amount of time by the standards of this talk, or by the standards of the numbers that appear in Ramsey's theorem. The numbers that appear in the finite Ramsey theorem are quite unusual for a fundamental theorem of combinatorics. As we shall review, exponential stacks of arbitrary length appear in the upper and lower bounds. We published a paper in the mid 80's which gave another related decision problem surrounding Ramsey's paper (the spectra comparison problem), where we show that the time complexity has upper and lower bounds involving exponential stacks of arbitrary length just as in Ramsey's theorem. H. Friedman, On the Spectra of Universal Relational Sentences, Information and Control, vol. 62, nos. 2/3, August/September 1984, pp. 205-209. We have another simple decision problem about finite structures with these same huge upper and lower bounds. We also have extended Ramsey's original logic problem to a more general context, and used Ramsey's theorem in a more exotic way to solve it. Again, the decision procedure is just nondeterministic exponential, but the numbers involved are far more gigantic. The Ackermann function is a mere speck of dust compared to the numbers involved here. This involves what is called the < 0 -hierarchy of numerical functions. We then give a decision problem in this more general context, which is analogous to the one published in the mid 80's, and
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2 give upper and lower bounds involving the < 0 -hierarchy of numerical functions. Thirdly, we give a decision problem in this more general context, which is analogous to the new decision problem about finite structures. We also establish upper and lower bounds involving the < 0 -hierarchy of numerical functions. There is a related problem that we touch on. By the completeness theorem, every valid sentence in predicate calculus has a proof in predicate calculus. However, the proof may be very long. In fact, there may well be a reasonably short proof that a small sentence in predicate calculus is valid, but all proofs in predicate calculus are gigantic. We give some extreme examples of this related to exotic Ramsey theory.
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