Ramsey Theory

Ramsey theory is a branch of mathematics studying the conditions under which some degree of order or structured pattern must necessarily appear within any sufficiently large mathematical or combinatorial system, however randomly or adversarially that system is arranged. British mathematician Frank Ramsey established the field's foundational result in a 1930 paper, proving what is now called Ramsey's theorem, which guarantees that in any sufficiently large complete graph with its edges colored using a fixed number of colors, a monochromatic complete subgraph of a specified size must exist. A commonly cited illustration of the theorem's flavor is the statement that among any group of six people, there must exist either three people who all mutually know each other or three who are all mutual strangers, a specific case of the general theorem. Ramsey theory has since grown into an active area of combinatorics with applications reaching into computer science, number theory and logic, and it is well known for typical results that guarantee some ordered structure must exist while the actual minimum size needed, called a Ramsey number, is often extremely difficult to determine exactly even for small cases.