Ramsey Theory

Ramsey Theory
Old, New and Unknown
Thiago Barros
Origins
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Frank Plumpton Ramsey
“Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe
that a bridge would fall down?”
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Born in 1903, Ramsey was the oldest
son of Arthur Ramsey, then President
of Magdalene College, Cambridge.
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He entered the Winchester College in
1915 and from there he won a
scholarship to Trinity College (1920).
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Graduated as a Wrangler in the
Mathematical Tripos of 1923.
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In 1924, he was elected a fellow at King's College, being the second
person ever to be elected to a fellowship not having previously
studied at King's.
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While Ramsey was a lecturer in Mathematics, he produced works
in different areas, including Economics, Philosophy and Logic.
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In Economics, Ramsey wrote A contribution to the theory of
taxation and A mathematical theory of saving, both of which
led to new areas in the subject.
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In Philosophy, Ramsey wrote several works, including Universals,
Knowledge, Theories and General propositions and
causality.
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It was with On a problem of formal logic, read to the London
Mathematical Society in 13 December 1928, however, that Ramsey
gave his seminal contribution to the branch of combinatorics that
now bears his name.
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Combinatorics was used by Ramsey to deal with a special case of the
decision problem for the first-order predicate calculus.
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As a point of interest, it is now known that there exists a more direct
proof than the one given by Ramsey and that the general case of the
decision problem cannot be solved. D.H. Mellor noted★:
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Ramsey's enduring fame in mathematics ... rests on a theorem he
didn't need, proved in the course of trying to do something we
now know can't be done!
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Ramsey died at the age of 26, suffering from chronic liver
problems.
The First Problem
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Friends and Strangers
“Of three ordinary people, two must have the same sex.”
D.J. Kleitman
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Ramsey theory is concerned with the conditions under which
certain patterns emerge.
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Typically, results in this field establish that for sufficiently large
structures (e.g. sets), a desired pattern is inevitable.
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Perhaps the simplest non-trivial example of a Ramsey problem is
the following “puzzle problem”.
Claim:
In any collection of six people, either three of them mutually
know each other or three of them mutually do not know each
other.
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Consider any group of six people, say, Professor Tom Brown, Dr.
Veselin Jungic, Moe, Helen, Professor Peter Borwein and Frank
Ramsey.
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Fix one person, say Moe, and consider his relation to the other 5
people.
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By the Pigeon-Hole principle, Moe either knows at least three
people or he does not know at least three.
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Suppose Moe knows Helen, Professor Brown and Professor Borwein
- if he doesn't know at least three people, we have an identical
argument.
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If some pair of these three know each other, say Professor Brown
and Helen, we are done - Moe, Helen and Professor Brown
mutually know each other.
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If no pair of the three know each other, then three mutually do not
know each other and, once again, we are done.
?
?
Or
“Knowing”
“Not-knowing”
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“Complete Disorder is
Impossible”
T.S. Motzkin
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Ramsey's Theorem Abridged
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The simple example involving six people raise a few interesting
points.
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If the mathematics department wishes to throw a party in which
either m people know each or n people do not know each other, is
there a number (R(m,n)) that will assure that this property will
always hold?
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We could translate the aforementioned question into graph theory
vocabulary and ask weather or not we can always find a graph
sufficiently large in which there always exists either a clique of
order m or an independent set of order n.
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The existence of R(m,n) is asserted by Ramsey's Theorem.
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A little bit of notation will help us at this point.
In effect, Ramsey proved a much more general version of the
“party” problem.
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References
Graham, R.L; Rothschild, B.L;
Spencer, J.H. Ramsey Theory, 2nd
edition, John Wiley & Sons.
★
✤
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Mellor, D.H. “The eponymous F. P.
Ramsey”. Journal of Graph
Theory 7 (1)(1983), 9-13.
Frank Plumpton Ramsey.
Retrieved from
www-groups.dcs.st-and.ac.uk/~his