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$s$-almost cross-$t$-intersecting families for vector spaces

Dehai Liu, Jinhua Wang, Tian Yao

TL;DR

The paper studies $s$-almost cross-$t$-intersecting families of $k$-dimensional subspaces in an $n$-dimensional vector space over $\mathbb{F}_{q}$, extending Erdős–Ko–Rado-type ideas to the vector-space setting. It develops a framework based on $t$-covers, Gaussian-binomial coefficient bounds, and auxiliary counting functions to bound the product $|\mathcal{F}|\,|\mathcal{G}|$ and to identify extremal structures. The main achievement is an exact extremal result: $|\mathcal{F}|\,|\mathcal{G}|\le {n-t\brack k-t}^{2}$ with equality precisely when $\mathcal{F}=\mathcal{G}=\mathcal{H}_{1}(V,E;k)$ for some fixed $E\in {V\brack t}$; the paper also provides stability formulations and refines the extremal configurations across regimes of $k$ relative to $t$, including explicit cross-$t$-intersecting constructions for certain cases.

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F} _{q} $, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are said to be cross-$t$-intersecting if $\dim(F\cap G)\ge t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result.

$s$-almost cross-$t$-intersecting families for vector spaces

TL;DR

The paper studies -almost cross--intersecting families of -dimensional subspaces in an -dimensional vector space over , extending Erdős–Ko–Rado-type ideas to the vector-space setting. It develops a framework based on -covers, Gaussian-binomial coefficient bounds, and auxiliary counting functions to bound the product and to identify extremal structures. The main achievement is an exact extremal result: with equality precisely when for some fixed ; the paper also provides stability formulations and refines the extremal configurations across regimes of relative to , including explicit cross--intersecting constructions for certain cases.

Abstract

Let be an -dimensional vector space over the finite field , and denote the family of all -dimensional subspaces of . The families are said to be cross--intersecting if for all . Two families and are called -almost cross--intersecting if each member of (resp. ) is -disjoint with at most members of (resp. ). In this paper, we discribe the structure of -almost cross--intersecting families with maximum product of their sizes. In addition, we prove a stability result.
Paper Structure (4 sections, 17 theorems, 102 equations)

This paper contains 4 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Inequalities concerning Gaussian binomial coefficients
  3. Proof of Theorem \ref{['dingli1']}
  4. Proof of Theorem \ref{['dingli5.2']}

Key Result

Theorem 1.1

Let $n$, $k$, $t$ and $s$ be positive integers with $k\geq t+1$ and $n\geq 2k+2t+1+\log_{q} 7s$. If $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are $s$-almost cross-$t$-intersecting families, then Equality holds if and only if $\mathcal{F}=\mathcal{G}=\left \{ H\in{V\brack k}: E\subseteq H \right \}$ for some $E\in {V\brack t}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 29 more