Finite Geometry and Ramsey Theory (25w5364)

The Banff International Research Station will host the “Finite geometry and Ramsey theory” workshop in Banff from September 7 to 12, 2025.

Ramsey theory is one of the central areas in discrete mathematics and has seen tremendous growth since its inception in the early 20th century. The goal of this area is to find order in chaos and the main mathematical principle behind it is the phenomenon that there are unavoidable patterns in every large enough structure. For example, one of the classical problems in this area can be stated as follows: for any number $k$, what is the smallest number $n$ such that in any group of $n$ people either there will be $k$ mutual friends or there will be $k$ mutual strangers? The existence of such a number $n$ was shown by Frank Ramsey in 1930s and while for $k = 3$ the smallest $n$ is known to be equal to $6$, the $k = 5$ case is already a major open problem in mathematics. This principle is truly far-reaching as it has been observed in mathematical disciplines as diverse as number theory, computer science, graph theory, logic and geometry. Finite geometry lies at the intersection of two fundamental areas of mathematics, geometry and discrete mathematics. It studies finite structures that satisfies some of the axioms that we are familiar with from classical geometry. This area also has deep ties with algebra, information theory and theoretical computer science.

Some of the recent breakthroughs in Ramsey theory have shown the importance of using finite geometric constructions. The main goal of this workshop is to deepen the connections between these two research areas, while developing new methods with wide applicability. We will initiate inter-disciplinary collaborations to tackle important open problems and train future experts to conduct ground-breaking research. Our workshop is also designed to be inclusive and diverse in terms of participation of early career researchers and underrepresented communities.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.