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RAMSEY THEORY AND ENORMOUS LOWER BOUNDS
Abstract
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
friedman@math.ohiostate.edu
www.math.ohiostate.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), 264286.
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. 205209.
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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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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 Fall '08
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