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$s$-almost $t$-intersecting families for finite sets

Dehai Liu, Kaishun Wang, Tian Yao

Abstract

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\left|\bigcap_{F\in\mathcal{F}} F\right|<t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.

$s$-almost $t$-intersecting families for finite sets

Abstract

A family of -subsets of an -set is called -almost -intersecting if each member is -disjoint with at most members. In this paper, we prove that, if is maximum, then consists of all -subsets containing a fixed -subset. Consequently, it is natural to consider the maximum-sized with . The famous Hilton-Milner theorem settles the case where is -intersecting. We characterize the remaining case completely.

Paper Structure

This paper contains 6 sections, 20 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['2509211']}
  4. Proof of Theorem \ref{['2509212']}
  5. Proof of Theorem \ref{['2509213']}
  6. Inequalities concerning binomial coefficients

Key Result

Theorem 1.1

Let $n$, $k$, $t$ and $s$ be positive integers with $k\geq t+1$ and $n\geq L_{1}(k,t,s)$. If $\mathcal{F}\subseteq \binom{[n]}{k}$ is an $s$-almost $t$-intersecting family with the maximum size, then there exists $X\in\binom{[n]}{t}$ such that $\mathcal{F}=\left\{F\in\binom{[n]}{k}: X\subseteq F\rig

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['2509211']}
  • ...and 41 more