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arXiv 1606.09575math.COmath.NT
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无向日葵集的上界

Upper bounds for sunflower-free sets

Eric Naslund, William F. Sawin

备注：
5 pages
发表信息：
Forum of Mathematics, Sigma, Volume 5, 2017, e15
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摘要：

一个$k$集合的集合被称为形成一个$k$ -sunflower或$Δ$ -system，如果集合中任意两个集合的交集相同，我们称一个集合族$\mathcal{F}$ sunflower-free如果它不包含sunflower。继最近的突破Ellenberg和Gijswijt和Croot，Lev和Pach，我们应用多项式方法直接到Erdens-Szemerédi向日葵问题，并证明了任何向日葵家庭$\mathcal{F}$的子集$\{1,2,\dots,n\}$的大小最多\[ |\mathcal{F}|\leq3n\sum_{k\leq n/3}\binom{n}{k}\leq\left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}. \]我们说，一个集$A\subset(\mathbb Z/D \mathbb Z)^{n}=\{1,2,\dots,D\}^{n}$的$D>2$是向日葵免费的，如果每个不同的三重$x,y,z\in A$存在一个坐标$i$，正好有两个$x_{i},y_{i},z_{i}$是相等的。使用一个版本的多项式方法与字符$χ:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}$，而不是多项式，我们表明，任何向日葵自由集$A\subset(\mathbb Z/D \mathbb Z)^{n}$有大小\[ |A|\leq c_{D}^{n} \]其中$c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$。这可以被看作是在证明Erdens-Rado向日葵猜想的可能方法上取得了进一步的进展，Alon，Sphilka和Umans的工作相当于证明$c_{D}\leq C$对于一些常数$C$独立于$D$。

英文摘要：

A collection of $k$ sets is said to form a $k$-sunflower, or $Δ$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ sunflower-free if it contains no sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach we apply the polynomial method directly to Erdős-Szemerédi sunflower problem and prove that any sunflower-free family $\mathcal{F}$ of subsets of $\{1,2,\dots,n\}$ has size at most \[ |\mathcal{F}|\leq3n\sum_{k\leq n/3}\binom{n}{k}\leq\left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}. \] We say that a set $A\subset(\mathbb Z/D \mathbb Z)^{n}=\{1,2,\dots,D\}^{n}$ for $D>2$ is sunflower-free if every distinct triple $x,y,z\in A$ there exists a coordinate $i$ where exactly two of $x_{i},y_{i},z_{i}$ are equal. Using a version of the polynomial method with characters $χ:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}$ instead of polynomials, we show that any sunflower-free set $A\subset(\mathbb Z/D \mathbb Z)^{n}$ has size \[ |A|\leq c_{D}^{n} \] where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$. This can be seen as making further progress on a possible approach to proving the Erdős-Rado sunflower conjecture, which by the work of Alon, Sphilka and Umans is equivalent to proving that $c_{D}\leq C$ for some constant $C$ independent of $D$.

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