Upper bounds for sunflower-free sets

A collection of \(k\) sets is said to form a \(k\)-sunflower, or \(\Delta\)-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-Szemer\'{e}di 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 \(\chi:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}\) 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 \(D\).