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Extremal problems for intersecting families of subspaces with a measure

Hajime Tanaka, Norihide Tokushige

TL;DR

Let $\Omega_n$ denote the subspace set of $\mathbb{F}_q^n$ and study the maximum $\mu_{\sigma}(U)$ of intersecting families under the $\sigma$-biased measure $\mu_{\sigma}$. The authors develop a $q$-analogue of the Erdős--Ko--Rado theory and prove that, for suitable $\sigma$, the maximum is $\frac{\sigma}{1+\sigma}$ attained by the 1-containing family $A_n^{(1)}$. They extend the framework to $t$-intersecting and cross $t$-intersecting families, deriving asymptotics for $f(n,t,\sigma_{\theta,n})$ and connecting to Frankl--Wilson and Cao--Lu--Lv--Wang results. The work combines semidefinite programming, spectral analysis, probabilistic concentration, and classic intersection bounds to provide a rigorous $q$-analogue of measure-EKR results and to relate them to Hoffman's bound and subset analogues.

Abstract

We introduce a measure for subspaces of a vector space over a $q$-element field, and propose some extremal problems for intersecting families. These are $q$-analogue of Erdős-Ko-Rado type problems, and we answer some of the basic questions.

Extremal problems for intersecting families of subspaces with a measure

TL;DR

Let denote the subspace set of and study the maximum of intersecting families under the -biased measure . The authors develop a -analogue of the Erdős--Ko--Rado theory and prove that, for suitable , the maximum is attained by the 1-containing family . They extend the framework to -intersecting and cross -intersecting families, deriving asymptotics for and connecting to Frankl--Wilson and Cao--Lu--Lv--Wang results. The work combines semidefinite programming, spectral analysis, probabilistic concentration, and classic intersection bounds to provide a rigorous -analogue of measure-EKR results and to relate them to Hoffman's bound and subset analogues.

Abstract

We introduce a measure for subspaces of a vector space over a -element field, and propose some extremal problems for intersecting families. These are -analogue of Erdős-Ko-Rado type problems, and we answer some of the basic questions.
Paper Structure (7 sections, 17 theorems, 144 equations, 1 table)

This paper contains 7 sections, 17 theorems, 144 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}
  4. Proof of Theorem \ref{['thm3']}
  5. Concluding remarks
  6. Theorem \ref{['thm1']} from Hoffman's bound
  7. Comparison of Theorem \ref{['thmG']} with the corresponding subset version

Key Result

Theorem A

Let $\frac{k}{n}\leqslant \frac{1}{2}$. If a family $U\subset\binom{X_n}{k}$ is intersecting, then Moreover, if $|U|/\binom{n}{k}=\frac{k}{n}$ and if $\frac{k}{n}< \frac{1}{2}$, then there exists $i\in X_n$ such that

Theorems & Definitions (32)

  • Theorem A: EKR
  • Theorem B: AK
  • Theorem C: Hsieh1975DM
  • Theorem 1
  • Conjecture 1
  • Theorem 2
  • Conjecture 2
  • Theorem 3
  • Lemma 1
  • proof
  • ...and 22 more