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Kleitman's conjecture for central families

Jonathan Cary

Abstract

Chvátal conjectured that a star is amongst the largest intersecting subfamiles of a finite subset-closed family of sets. Kleitman later strengthened Chvátal's conjecture, suggesting that maximal intersecting subfamilies of $2^{[n]}$ when naturally embedded into $\mathbb{R}^{2^{[n]}}$ take on a particular form. We provide a construction which succeeds in expressing certain families as required by Kleitman's conjecture. We then provide a partial characterization of these families, showing central and certain near-central families in particular to be amongst them.

Kleitman's conjecture for central families

Abstract

Chvátal conjectured that a star is amongst the largest intersecting subfamiles of a finite subset-closed family of sets. Kleitman later strengthened Chvátal's conjecture, suggesting that maximal intersecting subfamilies of when naturally embedded into take on a particular form. We provide a construction which succeeds in expressing certain families as required by Kleitman's conjecture. We then provide a partial characterization of these families, showing central and certain near-central families in particular to be amongst them.
Paper Structure (4 sections, 9 theorems, 28 equations)

This paper contains 4 sections, 9 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. A construction for Kleitman's conjecture
  3. Central families
  4. Conclusion

Key Result

Lemma 1.1

Conjecture kleitman implies chvatal.

Theorems & Definitions (28)

  • Conjecture 1.1
  • Definition 1.1
  • Conjecture 1.2
  • Lemma 1.1
  • proof
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Definition 2.3
  • ...and 18 more