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 $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge {\left(\frac{c{k}^2\ln m}{\ln \ln m}\right)}^m$ 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.