Sunflower Bound with a Sub-Logarithmic Base

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.

Minor errors in Ver. 1 fixed on 8 pages. Please open https://sites.psu.edu/sunflowerconjecture/2022/12/18/index-page/ for extra information such as proof details