Erdos-Ko-Rado problems in finite geometry, group theory and integer partitions

01 October 2021 → 31 December 2023
Research Foundation - Flanders (FWO)
Research disciplines
  • Natural sciences
    • Group theory and generalisations
    • Geometry
    • Combinatorics
Intersecting sets Integer partitions Permutation groups
Project description

This project will investigate Erdős-Ko-Rado (EKR) problems in different contexts. An EKR problem concerns large sets of pairwise intersecting objects. The definition of intersecting depends on the context. 1) EKR in vector spaces. We consider sets of k-spaces pairwise intersecting in a subspace of dimension contained in D={t_1,t_2,…,t_s}. The goal is to classify the largest examples. These sets are interesting from a coding theoretical point of view: they give rise to subspace codes, which are used in random network coding. 2) EKR in permutation groups. Let G be a finite group acting on a set X. Two elements A,B in G are intersecting if there exists an element i in X such that A(i)=B(i). For many permutation groups, the largest EKR sets are known. In this project I will look for the second largest maximal examples of EKR sets, starting with the projective linear groups PGL(n+1,q). Such a stability result will lead to new classification results for Cameron-Liebler sets. 3) EKR in integer partitions. Two integer partitions of a positive integer n are intersecting if they have a summand in common. If the length of the partition is fixed, then the largest intersecting set of integer partitions is the set of all integer partitions containing the summand one. For large values of n, this result is not true if we do not fix the length of the partitions. My goal is to solve the EKR problem for all sets of integer partitions of n, without restriction on the length.