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Odd-Sunflowers

Peter Frankl, János Pach, Dömötör Pálvölgyi

Abstract

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant $μ<2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most $μ^n$ sets. We construct such families of size at least $1.5021^n$. We also characterize minimal odd-sunflowers of triples.

Odd-Sunflowers

Abstract

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant such that every family of subsets of an -element set that contains no odd-sunflower consists of at most sets. We construct such families of size at least . We also characterize minimal odd-sunflowers of triples.
Paper Structure (8 sections, 12 theorems, 13 equations)

This paper contains 8 sections, 12 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:even']}
  3. Direct Sum Constructions
  4. Wreath Product Constructions
  5. Minimal odd-sunflowers (MOS-s)
  6. Concluding remarks
  7. $\mathbf{NP}$-hardness sketch
  8. Proof of Proposition \ref{['prop:3MOS']}

Key Result

Theorem 2

For any even-sunflower-free family $\mathcal{F}\subset 2^{\{1,\dots,n\}}$, we have $|\mathcal{F}|\le n$. That is,

Theorems & Definitions (26)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof : First proof
  • proof : Second proof.
  • Lemma 4
  • proof
  • Lemma 5
  • Remark
  • proof
  • ...and 16 more