Open problem, Combinatorics and discrete mathematics, posed 1960

Erdős–Rado sunflower conjecture

OpenErdős

For every integer $r \ge 3$ , there exists a constant $C_r > 0$ depending only on $r$ such that any family $\mathcal{F}$ of sets each of cardinality at most $w$ with $|\mathcal{F}| \ge C_r^w$ contains an $r$ -sunflower (or $\Delta$ -system)—that is, $r$ distinct sets $S_1, \dots, S_r \in \mathcal{F}$ whose pairwise intersections are all equal to a common core $K$ ( $S_i \cap S_j = K$ for all $i \neq j$ ).

Posed by, 1960Paul Erdős

Research frontier as of 2026

As of 2026, the best known upper bound for the $r$ -sunflower problem on $w$ -uniform families is $|\mathcal{F}| \ge (C r \log w)^w$ (Bell–Chueluecha–Warnke, 2021, following Alweiss–Lovett–Wu–Zhang, 2019 and Rao, 2020). This bound holds for the stronger notion of $(1/r, 1/r)$ -approximate (robust) sunflowers, for which the $\log w$ factor is actually tight. Eliminating the remaining $\log w$ factor to reach $C_r^w$ even for $r = 3$ therefore requires arguments that distinguish exact disjointness of petals from probabilistic robust disjointness.

Best known results

Every family of $w$ -element sets of size $|\mathcal{F}| \ge (C r \log w)^w$ contains an $r$ -sunflower (Alweiss–Lovett–Wu–Zhang, 2019; Rao, 2020; Bell–Chueluecha–Warnke, 2021).
The weak sunflower conjecture in $\{0, 1\}^n$ (or a sunflower in $\mathbb{F}_3^n$ ) is proved with an exponential bound $c^n$ ( $c < 2$ ) via the slice-rank polynomial method (Naslund and Sawin, 2017).

Tools and where they stop

Tool	Achieved	Where it stops
κ-spread set systems and Shannon entropy encoding	Reduces general families to $\kappa$ -spread families (where no nonempty subset is contained in more than $\kappa^{-\|T\|} \|\mathcal{F}\|$ sets) and shows that a random ground set of density $\frac{1}{2r}$ contains a member of $\mathcal{F}$ with probability near $1$ , yielding $(C r \log w)^w$ .	A random set of density $\frac{1}{2r}$ only covers a $w$ -set with high probability when the spread parameter satisfies $\kappa = \Omega(r \log w)$ , making the $\log w$ factor unavoidable for robust sunflowers.
Slice-rank polynomial method	Proves exponential bounds $(3 / 2^{2/3})^n \approx 1.8899^n$ for sunflower-free subsets of $\{0, 1\}^n$ when the ground set size $n$ is fixed.	Bounds depend exponentially on the ambient universe size $n$ rather than the set size $w$ , failing when $n \gg w$ .

Open questions

Does there exist an absolute constant $C_3 > 0$ such that every family of $w$ -element sets of size at least $C_3^w$ contains a $3$ -sunflower?
Can the $(C r \log w)^w$ bound be improved to $C_r^w (\log w)^{o(w)}$ for fixed $r = 3$ ?

References

Paul Erdős, Richard Rado (1960). Intersection theorems for systems of sets · DOI:10.1112/jlms/s1-35.1.85
Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang (2021). Improved bounds for the sunflower lemma · DOI:10.4007/annals.2021.194.3.5 · arXiv:1908.08483