Higher-rank characterization of the local weak-EKR condition

Characterize the irreducible linear groups L ≀ GL(V), for V = 𝔽_p^d with d β‰₯ 3, such that every subspace W ≀ V and every subset A βŠ† V satisfying A βˆ’ A βŠ† ⋃_{β„“βˆˆL} W^β„“ also satisfies |A| ≀ |W|.

Background

The paper characterizes finite groups with the weak ErdΕ‘s–Ko–Rado property by imposing a local intersection-density condition on each chief factor and the linear group induced on that factor. For chief factors of rank two, Proposition \ref{prop: rank 2} proves that the condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces.

The unresolved extension concerns elementary abelian chief factors of rank at least three. Given an irreducible linear group L acting on V = 𝔽_pd, the problem asks which groups ensure that a subset whose pairwise differences lie in the union of the L-conjugates of every prescribed subspace W cannot exceed |W| in size. A solution would provide the higher-rank analogue of the complete rank-two characterization and would clarify the local criterion used in the characterization of weak EKR groups.

References

We conclude this section with an open problem. Proposition~\ref{prop: rank 2} gives a complete characterization of the condition appearing in Theorem~\ref{thm:chief-series-characterization} when the chief factor has rank two. It would be interesting to obtain an analogous characterization in higher rank.

\begin{problem}\label{prob:higher-rank} Let $V=\mathbb{F}_pd$, where $d\geq 3$, and let $L\leq GL(V)$ be an irreducible linear group. Characterize those groups $L$ for which

A-A\subseteq \bigcup_{\ell\in L}W\ell \quad\Longrightarrow\quad |A|\leq |W|

for every subspace $W\leq V$ and every subset $A\subseteq V$. \end{problem}

— Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers  (2608.18594 - Frelih et al., 19 Aug 2026) in Section 2, immediately preceding Problem 2.1 (labeled prob:higher-rank)