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A Relationship for LYM Inequalities between Boolean Lattices and Linear Lattices with Applications

Jiuqiang Liu, Guihai Yu

Abstract

Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no $s$ pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no $d$-dimensional Boolean algebras.

A Relationship for LYM Inequalities between Boolean Lattices and Linear Lattices with Applications

Abstract

Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no -dimensional Boolean algebras.
Paper Structure (11 sections, 41 theorems, 140 equations)

This paper contains 11 sections, 41 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems 1.9 and 1.11
  3. Sperner families and Diamond-free families
  4. Sperner and $k$-Sperner families
  5. Diamond-free families
  6. Sharpening the LYM inequality
  7. Families with no s pairwise disjoint members
  8. Preliminaries
  9. Proofs of Theorems 1.13 and 1.14
  10. Families without $d$-dimensional $q$-algebras in linear lattices
  11. Concluding Remarks

Key Result

Theorem 1.1

(Sperner Theorem). Suppose that $\mathcal{A}$ is a Sperner family of subsets of $[n]$. Then

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 40 more