From algebraic graph theory to finite geometry to extremal combinatorics and back

Project Details

Description

This project will strengthen the interaction between three adjacent
domains of research: finite geometry, extremal combinatorics and
algebraic graph theory. Recent advances have shown the exciting
potential of this cross pollination, but often lack the background to
fulfill this promise. I aim to bridge this gap by making progress on
some important open problems along two main themes.
(A) I will investigate constructions of H-free graphs from finite
geometry in relation to Ramsey and Turan numbers. I will use
advanced ideas from finite geometry, some of which will be applied
for the first time to this topic. The ambitious goals are to improve
bounds on off-diagonal Ramsey numbers and refute an old
conjecture on girth 5 graphs.
(B) The intimate relation between classical groups and finite
geometry can be understood via buildings and their association
schemes. We recently proved a result on subsets of elements that
are "not far away" by examining these algebras, generalizing and
simplifying previous work. I will continue this project, as many
questions remain. I will further immerse myself in the theory of
buildings, combine it with my current expertise and make progress on
these questions. The same techniques are applicable to association
schemes from permutation groups where similar questions have
recently gained traction, but are not fully resolved.
This proposal will allow me to expand my expertise in a variety of
topics and establish myself as a well-rounded researcher.
AcronymFWOTM1106
StatusNot started
Effective start/end date1/10/2230/09/25

Keywords

  • finite geometry
  • extremal graph theory
  • association schemes

Flemish discipline codes

  • General mathematics