  A sunflower with $k$ petals, or $k$-sunflower, is a family of $k$ sets every
two of which have a common intersection. Known since 1960, the sunflower
conjecture states that a family ${\mathcal F}$ of sets each of cardinality $m$
includes a $k$-sunflower if $|{\mathcal F}| \ge c_k^m$ for some $c_k \in
{\mathbb R}_{>0}$ depending only on $k$. The case $k=3$ of the conjecture was
especially emphasized by Erd\"os, for which Kostochka's bound $c m! \left(
\frac{\log \log \log m}{\log \log m} \right)^m$ on $|{\mathcal F}|$ without a
3-sunflower had been the best-known since 1997 until the recent development to
update it to $c \log m$. This paper proves with an entirely different
combinatorial approach that ${\mathcal F}$ includes three mutually disjoint
sets if it satisfies the $\Gamma \left( c m^{\frac{1}{2}+ \delta}
\right)$-condition for any given $\delta \in (0, 1/2)$. Here $c$ is a constant
depending only on $\delta$, and the $\Gamma$-condition refers to \[ | \left\{
U~:~ U \in {\mathcal F} \textrm{~and~} S \subset U \right\}| < \left( c
m^{\frac{1}{2}+ \delta} \right)^{-|S|} |{\mathcal F}|, \] for every nonempty
set $S$. This poses an alternative proof of the 3-sunflower bound $\left( c
m^{\frac{1}{2}+ \delta} \right)^m$.
