向日葵界的渐近改进
Asymptotic Improvement of the Sunflower Bound
摘要：
具有核心$Y$的向日葵是${\cal B}$家族的集合，使得$U \cap U' = Y$对于${\cal B}$中的每两个不同元素$U$和$U'$。著名的向日葵引理指出，一个给定的集合族${\cal F}$，每个集合的基数至多为$s$，如果$|{\cal F}|> (k-1)^s s!$，则包含一个基数为$k$的向日葵。自从Erdös和Rado在1960年证明了这一点以来，半个多世纪以来，人们一直不知道向日葵束缚的$(k-1)^s s!$是否可以对任何$k$和$s$进行渐近改进。证明了对于某个实数$c_k>0$，它可以归结为$c_k^s$，这就是向日葵猜想。本文表明，一般的向日葵界确实可以减少指数因子：我们证明，${\cal F}$包括向日葵的基数$k$，如果\[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \]为常数$c>0$，和任何$k \ge 2$和$s \ge 2$。例如，当给定恒定$ε\in (0,1)$的$k \ge s^ε$时，向日葵结合从$(k-1)^s s!$减少到$(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$，实现减少比率$\left[ O \left( \frac{1}{\log s} \right) \right]^s$。基数至少为$\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$的任何${\cal F}$也包括基数为$k$的向日葵，其中$\frac{1}{\sqrt{10}-2}=0.8603796\ldots$。我们的研究结果表明，向日葵绑定可以提高一个因素，小于一个小常数的权力$s$，希望进一步更新。
英文摘要：
A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. Since Erdös 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 ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$. For instance, whenever $k \ge s^ε$ for a given constant $ε\in (0,1)$, the sunflower bound is reduced from $(k-1)^s s!$ to $(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$, achieving the reduction ratio of $\left[ O \left( \frac{1}{\log s} \right) \right]^s$. Also any ${\cal F}$ of cardinality at least $\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\ldots$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
