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Extensions of a Family for Sunflowers

Junichiro Fukuyama

TL;DR

The paper develops a framework for the combinatorial domain 2^X to study sunflowers via extension generators. It introduces weighted norms and Gamma-conditions, proves a collection of extension- and growth-lemmas, and employs partitioning of X into structured strips to enable an inductive construction of disjoint sets. The central contributions include the Extension Generator Theorem and a strong KDC theorem that guarantees the existence of k mutually disjoint sets under a parameterized Gamma condition, thereby strengthening sunflower-type results. These methods connect norm-based analysis, combinatorial partitioning, and extension-generating techniques to advance extremal set theory and delta-system phenomena with potential implications for sunflower conjectures.

Abstract

This paper explores the structure of the combinatorial domain $2^X$ in relation to sunflowers. The previous study found some intrinsic properties of the $l$-extension \[ Ext \left( \mathcal{F}, l \right) = \left\{ V ~:~ V \in {X \choose l},~ \exists U \in \mathcal{F}~ U \subset V \right\} \] of a family $\mathcal{F}$ of $m$-cardinality sets. Subsequently, it lead to the proof that such an $\mathcal{F}$ includes three mutually disjoint sets if it satisfies the $Γ(b)$-condition, that is, \[ \left| \mathcal{F}[S] \right| < b^{-|S|} |\mathcal{F}| \quad \textrm{for every nonempty set}~ S, \qquad \textrm{where} \quad \mathcal{F} [S] := \left\{ U : U \in \mathcal{F},~ S \subset U \right\}, \] for $b= m^{\frac{1}{2}+ ε}$ with an $m$ sufficiently larger than a given constant $1/ε$. It is stronger than the statement that $\mathcal{F}$ includes a 3-sunflower if $|\mathcal{F}| > b^m$, where $k$-sunflower refers to a family of $k$ different sets with a common pair-wise intersection. Further refining the theory, we show that an $\mathcal{F}$ includes $k$ mutually disjoint sets if it satisfies the $Γ\left( 8^{\sqrt{\log_2 m}} \sqrt m ~k \log_2 k \right)$-condition with an $m$ sufficiently larger than $k$.

Extensions of a Family for Sunflowers

TL;DR

The paper develops a framework for the combinatorial domain 2^X to study sunflowers via extension generators. It introduces weighted norms and Gamma-conditions, proves a collection of extension- and growth-lemmas, and employs partitioning of X into structured strips to enable an inductive construction of disjoint sets. The central contributions include the Extension Generator Theorem and a strong KDC theorem that guarantees the existence of k mutually disjoint sets under a parameterized Gamma condition, thereby strengthening sunflower-type results. These methods connect norm-based analysis, combinatorial partitioning, and extension-generating techniques to advance extremal set theory and delta-system phenomena with potential implications for sunflower conjectures.

Abstract

This paper explores the structure of the combinatorial domain in relation to sunflowers. The previous study found some intrinsic properties of the -extension of a family of -cardinality sets. Subsequently, it lead to the proof that such an includes three mutually disjoint sets if it satisfies the -condition, that is, \[ \left| \mathcal{F}[S] \right| < b^{-|S|} |\mathcal{F}| \quad \textrm{for every nonempty set}~ S, \qquad \textrm{where} \quad \mathcal{F} [S] := \left\{ U : U \in \mathcal{F},~ S \subset U \right\}, \] for with an sufficiently larger than a given constant . It is stronger than the statement that includes a 3-sunflower if , where -sunflower refers to a family of different sets with a common pair-wise intersection. Further refining the theory, we show that an includes mutually disjoint sets if it satisfies the -condition with an sufficiently larger than .
Paper Structure (13 sections, 18 theorems, 105 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On the Combinatorial Domain $2^X$
  3. Relations between Sets
  4. Primitive Weight on $X$
  5. Collective $\Gamma$-Conditions
  6. Partitioning $X$ by a Given ${\mathcal{F}}$
  7. Proof of Theorem \ref{['KDC']}
  8. Proof of Some Statements in Sections 1 and 2
  9. Relations in $\left( 2^X \right) ^g$ and $\Gamma$-Conditions
  10. Proof of Theorem \ref{['EGT42']}
  11. Confirmation of the Extension Generator Theorem
  12. About Splits of $X$
  13. Asymptotics of Binomial Coefficients

Key Result

Theorem 1.1

There exists $c \in {\mathbb R}_{>1}$ such that for every $k \in {\mathbb Z}_{>1}$ and $m \in \left( 2^{ck} , \infty\right) \cap {\mathbb Z}$, a family ${\mathcal{F}}$ of sets each of cardinality $m$ includes $k$ mutually disjoint sets if it satisfies the $\Gamma(b)$-condition, where

Figures (1)

  • Figure 1: Algorithm GrowCores to Further Update $C_i$ and ${\mathcal{F}}_i$

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Corollary 2.8
  • ...and 18 more