# Ramsey Theory

## Transcription

Ramsey Theory

Ramsey Theory Old, New and Unknown Thiago Barros Origins 2 3 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?” • Born in 1903, Ramsey was the oldest son of Arthur Ramsey, then President of Magdalene College, Cambridge. • He entered the Winchester College in 1915 and from there he won a scholarship to Trinity College (1920). • Graduated as a Wrangler in the Mathematical Tripos of 1923. 4 • 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. • While Ramsey was a lecturer in Mathematics, he produced works in different areas, including Economics, Philosophy and Logic. • 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. • In Philosophy, Ramsey wrote several works, including Universals, Knowledge, Theories and General propositions and causality. • 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. • Combinatorics was used by Ramsey to deal with a special case of the decision problem for the first-order predicate calculus. • 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★: 5 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! • Ramsey died at the age of 26, suffering from chronic liver problems. The First Problem 6 7 Friends and Strangers “Of three ordinary people, two must have the same sex.” D.J. Kleitman • Ramsey theory is concerned with the conditions under which certain patterns emerge. • Typically, results in this field establish that for sufficiently large structures (e.g. sets), a desired pattern is inevitable. • 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. 8 Consider any group of six people, say, Professor Tom Brown, Dr. Veselin Jungic, Moe, Helen, Professor Peter Borwein and Frank Ramsey. 9 10 • Fix one person, say Moe, and consider his relation to the other 5 people. • By the Pigeon-Hole principle, Moe either knows at least three people or he does not know at least three. • Suppose Moe knows Helen, Professor Brown and Professor Borwein - if he doesn't know at least three people, we have an identical argument. • 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. • 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” 11 “Complete Disorder is Impossible” T.S. Motzkin 12 13 Ramsey's Theorem Abridged • The simple example involving six people raise a few interesting points. • 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? • 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. 14 • • The existence of R(m,n) is asserted by Ramsey's Theorem. • A little bit of notation will help us at this point. In effect, Ramsey proved a much more general version of the “party” problem. 15 References Graham, R.L; Rothschild, B.L; Spencer, J.H. Ramsey Theory, 2nd edition, John Wiley & Sons. ★ ✤ 16 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