A Survey of Ramsey Theory
Date of Graduation
Master of Science in Mathematics
Given any collection of six people at a party, three of them must either be mutual acquaintances or mutual strangers. Suppose one had a larger gathering. Ramsey theory proves that certain smaller collection of mutual acquaintances or strangers must always exist. (For instance, at any gathering of nine people there must either be three mutual acquaintances or four mutual strangers.) More generally, Ramsey proves the existence of such smaller groups when there are an arbitrary number of possible relationships. Although it is fairly simple to prove such existence, it is very difficult to determine the exact number of people that must be present to obtain certain smaller groups of mutual relationship. The purpose of this paper is to prove Ramsey's abridged and unabridged theorems, demonstrate the relationship between Ramsey's theorems and Van der Waerden's theorem, and to present the exact Ramsey numbers and bounding formulas that are currently known.
© Michele Bilton
Bilton, Michele, "A Survey of Ramsey Theory" (2001). MSU Graduate Theses. 2431.