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Source. Theorem 5, p. 2, with its proof in Section 3, pp. 4--5, of Eric Naslund and William F. Sawin, Upper bounds for sunflower-free sets, Forum Math. Sigma 5 (2017), Paper No. e15, doi:10.1017/fms.2017.12. Labels and pages here are those of arXiv:1606.09575v1, the edition named on the source card.

Statement

Definition (p. 2, after Alon, Shpilka and Umans, Definition 2.5). For $k\le D$, a $k$-sunflower in $(\mathbb Z/D\mathbb Z)^n$ is a set of $k$ vectors that in each coordinate are either all different or all the same. A set $A\subset(\mathbb Z/D\mathbb Z)^n$ is sunflower-free when it contains no 3-sunflower; the abstract (p. 1) phrases this as: every triple of distinct $x,y,z\in A$ has a coordinate $i$ in which exactly two of $x_i,y_i,z_i$ are equal.

Theorem 5 (p. 2, quoted). "Let $D\geq 3$, and let $A\subset(\mathbb{Z}/D\mathbb{Z})^n$ be a sunflower-free set. Then

$$|A|\leq c_D^n$$

where $c_D=\frac{3}{2^{2/3}}(D-1)^{2/3}$."

Proof pointer

Section 3, pp. 4--5. The proof replaces polynomials by the characters of $\mathbb Z/D\mathbb Z$. By orthogonality, a product over coordinates of character sums gives a function $T(x,y,z)$, displayed as (3.1) on p. 5, that is nonzero exactly when $x,y,z$ form a sunflower or are all equal; on a sunflower-free $A$ it is diagonal, so Lemma 6 (p. 2) bounds $|A|$ by its slice rank. Each term of the expansion has at most $2n$ nontrivial characters, so one of the three variables carries at most $2n/3$ of them; grouping by that variable bounds the slice rank by $3\sum_{k\le2n/3}\binom nk(D-1)^k\le3c_D^n$, using $D\ge3$. A tensor-power amplification removes the factor $3$.

Read depth

Claims checked: the definition, the statement and the proof outline were read on the print. Nothing here is independently reviewed.

Dependencies

Lemma 6 (p. 2), quoted by the paper from Tao. Nothing in the corpus.

Bears on

• Problem 20: by Alon, Shpilka and Umans (Theorem 2.6), a bound $C^n$ with $C$ independent of $D$ for 3-sunflower-free sets in $(\mathbb Z/D\mathbb Z)^n$ would give the problem's bound for $k=3$ with $c_3=e\cdot C$ (p. 2). The paper calls Theorem 5 progress towards that conjecture, but its $c_D$ grows like $D^{2/3}$, so it gives no bound independent of $D$ and proves no case of the problem.