Sunflower Bound with a Sub-Logarithmic Base

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.