IRMACS Ramsey Theory Working Group

Award Month: 
January - March, 2011

The Ramsey Theory Working Group at the IRMACS Centre was established in 2005. The group is generally interested in the Ramsey type problems, although it discusses any problem that a group member finds interesting. The group members meet once a week to talk about new papers, new and old problems and ideas, or to just enjoy being together and share their passion for mathematics.

Ramsey Theory is a cohesive sub-discipline of combinatorics. The theme of Ramsey Theory is that ‘complete chaos is impossible.’ Or, one could say that Ramsey Theory is ‘the study of unavoidable regularities in large structures.’

There are applications of Ramsey Theory to number theory, geometry, topology, set theory, logic, ergodic theory, information theory, and theoretical computer science. (A recent survey article on applications of Ramsey Theory has a bibliography of 252 items.)

Two of the largest branches of Ramsey Theory start with either ‘Ramsey's Theorem’ on the one hand, or ‘van der Waerden's Theorem on Arithmetic Progressions’ on the other. These two branches sometimes overlap, but a great number of results can be placed on one branch or the other.

A few examples of various problems in Ramsey theory that the IRMACS Ramsey Theory Working Group has been discussing, both new and longstanding, follow:

- Is it true that any sequence with at least four symbols, i.e., any sequence a1, a2,... with ai  S, |S|≥ 4, contains two blocks of the same sum and same length? 
- Is it true that any permutation of positive integers contains a monotone 4-term arithmetic progression? 
- Is it true that every 2-large set is large?

Quite recently the group was able to answer the longstanding question by Erdos and Graham if every linear ordering of the set of real numbers contains long monotone arithmetic progressions.

Current Member Listing: Hayri Ardal, Tom Brown, and Veselin Jungic.

About Project Leader: Hayri Ardal, Tom Brown, and Veselin Jungic