3-cluster-free families of subspaces

Gabriel Currier¹ ; Shahriar Shahriari²

¹University of British Columbia
²Pomona College

The electronic journal of combinatorics, Tome 32 (2025) no. 2

Résumé

Three $k$-dimensional subspaces $A$, $B$, and $C$ of an $n$-dimensional vector space $V$ over a finite field are called a $3$-cluster if $A \cap B \cap C = \{\mathbf{0}_V\}$ and yet $\dim(A+B+C) \leq 2k$. A special kind of $3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$ such that $A = (A \cap B )\oplus (A \cap C)$. We prove that, for $2 \leq k \le n/2$, the largest size of a covering triple-free family of $k$-dimensional subspaces is the same as the size of the largest such star (a family of subspaces all containing a designated non-zero vector). Moreover, we show that if $k < n/2$, then stars are the only families achieving this largest size. This in turn implies the same result for $3$-clusters, which gives the vector space-analogue of a theorem of Mubayi for set systems.

Détail

DOI : 10.37236/12962

Classification : 05D05, 05C35
Mots-clés : set systems, nontrivial family, Erdős-Ko-Rado theorem