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Sunflower Bound with a Sub-Logarithmic Base

Junichiro Fukuyama

TL;DR

The paper advances sunflower theory by proving that a family of $m$-sets contains a $k$-sunflower whenever $|\mathcal{F}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right)^m$, for some $c>0$. It introduces a $\Gamma(b)$-condition framework and a multi-step constructive method to extract a core and build a sunflower, achieving a sub-logarithmic base in $m$ for the exponential bound. This improves prior results (notably for $k=3$) and contributes toward the sunflower conjecture by providing a scalable path to tighter bounds. The approach blends combinatorial decomposition, probabilistic conditioning, and core-driven reduction to deliver a near-optimal exponential growth rate with respect to $m$.

Abstract

We show that a family $\mathcal{F}$ of sets each of cardinality $m \in \mathbb{Z}_{>2}$ includes a $k$-sunflower if $ |\mathcal{F}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right)^m$ for some constant $c>0$, where $k$-sunflower means a family of $k$ different sets with a common pairwise intersection. The base of the exponential lower bound is sub-logarithmic for each $k$ updating the current best-known result.

Sunflower Bound with a Sub-Logarithmic Base

TL;DR

The paper advances sunflower theory by proving that a family of -sets contains a -sunflower whenever , for some . It introduces a -condition framework and a multi-step constructive method to extract a core and build a sunflower, achieving a sub-logarithmic base in for the exponential bound. This improves prior results (notably for ) and contributes toward the sunflower conjecture by providing a scalable path to tighter bounds. The approach blends combinatorial decomposition, probabilistic conditioning, and core-driven reduction to deliver a near-optimal exponential growth rate with respect to .

Abstract

We show that a family of sets each of cardinality includes a -sunflower if for some constant , where -sunflower means a family of different sets with a common pairwise intersection. The base of the exponential lower bound is sub-logarithmic for each updating the current best-known result.
Paper Structure (2 sections, 3 theorems, 6 equations)

This paper contains 2 sections, 3 theorems, 6 equations.

Table of Contents

  1. Motivation, Basic Terminology and Related Facts
  2. Proof of Theorem \ref{['KSF']}

Key Result

Theorem 1.1

There exists $c \in {\mathbb R}_{>0}$ such that for every $k, m \in {\mathbb Z}_{>2}$, a family ${\mathcal{F}}$ of sets each of cardinality $m$ includes a $k$-sunflower if $|{\mathcal{F}}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right) ^m$. ∎

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof