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$3$-cluster-free families of subspaces

Gabriel Currier, Shahriar Shahriari

Abstract

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.

$3$-cluster-free families of subspaces

Abstract

Three -dimensional subspaces , , and of an -dimensional vector space over a finite field are called a -cluster if and yet . A special kind of -cluster, which we call a covering triple, consists of subspaces such that . We prove that, for , the largest size of a covering triple-free family of -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 , then stars are the only families achieving this largest size. This in turn implies the same result for -clusters, which gives the vector space-analogue of a theorem of Mubayi for set systems.
Paper Structure (5 sections, 12 theorems, 19 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Idea of the Proof for the Main Theorem
  4. Proof of Main Theorem
  5. Concluding remarks and further directions

Key Result

Theorem 1

Fix integers $k$ and $n$ with $n \geq 2k$, and suppose $\mathcal{F}$ is an intersecting family of $k$-dimensional subspaces of an $n$-dimensional vector space over a finite field. Then $\left\lvert {\mathcal{F}} \right\rvert$ is no more than the size of the largest such star. Moreover, for $n > 2k$,

Figures (2)

  • Figure 1: $A$ and $B$ are $k$-dimensional subspaces of the $n$-dimensional space $V$. $A \cap B$ is an $i$-dimensional subspace containing $\langle v \rangle$, and $(A+B)/A$ is a $k-i$ dimensional subspace of the $(n-k)$-dimensional space $V/A$.
  • Figure 2: $\dim(B \cap C) = i_A(B)$. $C$ could be $A$ or not, but, regardless, $D \cap A = \{\mathbf{0}_V\}$ for all $D \in \phi_A(B)$

Theorems & Definitions (25)

  • Theorem 1: Hsieh 1975 Hsieh:75, Frankl-Wilson 1986 FranklWilson:86, Godsil-Newman 2006 GodsilNewman:06, Chowdhury-Patkós 2010 ChowdhuryPatkos:10
  • Lemma 2
  • Theorem 3: Gerbner and Patkós GerbnerPatkos:09
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Lemma 7
  • proof
  • ...and 15 more