Asymptotic Improvement of the Sunflower Bound

A sunflower with a core $Y$ is a family $\mathcal{B}$ of sets such that $U \cap U’ = Y$foreachtwodifferentelements$U$and$U’$in${\cal B}$. The well-known sunflower lemma states that a given family $\mathcal{F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. Since Erd"os and Rado proved it in 1960, it has not been known for more than half a century whether the sunflower bound $(k-1{)}^{s}s!$ can be improved asymptotically for any $k$ and $s$. It is conjectured that it can be reduced to $c_k^s$ for some real number $c_k>0$ depending only on $k$, which is called the sunflower conjecture. This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that $\mathcal{F}$ includes a sunflower of cardinality $k$ if $$\mathrm{\mid }\mathcal{F}\mathcal{\mid }\ge {(\sqrt{10}-2)}^2{[k\cdot \min (\frac{1}{\sqrt{10}-2},\frac{c}{\log \min (k,s)})]}^{s}s!,$$ for a constant $c>0$, and any $k\ge 2$ and $s\ge 2$. For instance, whenever $k \ge s^\epsilon$foragivenconstant$\epsilon \in (0,1)$, the sunflower bound is reduced from $(k-1{)}^{s}s!$ to $(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$,achievingthereductionratioof$\left[ O \left( \frac{1}{\log s} \right) \right]^s$\mathrm{.}Alsoany${\cal F}$ of cardinality at least ${(\sqrt{10}-2)}^2{\left(\frac{k}{\sqrt{10}-2}\right)}^{s}s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\dots$. Our result demonstrates that the sunflower bound can be improved by a factor of less than a small constant to the power $s$, giving hope for further update.