Sunflower Bound with a Sub-Logarithmic Base

TL;DR

This paper establishes a new lower bound for the size of set families that guarantees the existence of a sunflower with a sub-logarithmic base, improving upon previous bounds in combinatorics.

Contribution

It introduces a novel exponential lower bound with a sub-logarithmic base for the sunflower conjecture, advancing theoretical understanding in combinatorics.

Findings

New lower bound for sunflower existence in set families

Bound has sub-logarithmic base, smaller than previous results

Advances theoretical limits in combinatorial set theory

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.