Projective geometry in (extremal) combinatorics

Unfortunately too many mathematicians nowadays consider (finite) projective geometry as a niche area of combinatorics. A reason for this may be that the focus of a lot of research in this area is on sometimes extremely specialized topics that are not necessarily of immediate interest to the broader combinatorial community, or at least are perceived to be such by that community.

However, projective geometry is not only interesting in its own right, it also has many links with hot and important topics in various other areas of combinatorics. In this talk I want to focus on problems in (extremal) combinatorics that are interesting to a broad (combinatorial) audience and involve, or have close relations to, projective geometry. I will discuss both some classical results, recent new work, and possibilities for future research. I will mostly focus on Turán-type problems related to \(C_k\)-free graphs, and problems related to certain types of independence sets in bipartite graphs.

It is my hope that this talk will show young people in projective geometry that they are not working in a niche area, and will motivate them to outreach to the broader combinatorial community convinced that projective geometry can provide an important and interesting contribution.