Erdős–Rado sunflower conjecture

Open

erdos-sunflower

Statement

A $k$ -sunflower is a family of $k$ sets with a common pairwise intersection (core). Prove that any family of more than $C_{k}^{w}$ sets of size $w$ contains a $k$ -sunflower, for a constant $C_{k}$ depending only on $k$ .

Current frontier

erdosproblems.com/20, OPEN. Erdős–Rado (1960) gave $\approx w! (k - 1)^{w}$ ; Alweiss–Lovett–Wu–Zhang (2020) and refinements reached $(C k lo g w)^{w}$ . The conjectured $C_{k}^{w}$ (no $lo g w$ in the base) is open, including $k = 3$ .

When this counts as solved

QUANTITATIVE

PROOF_COMPLETE for the $C_{k}^{w}$ bound, COUNTEREXAMPLE for faster growth. BREAKTHROUGH for removing the residual $lo g w$ factor for general $k$ , or settling $k = 3$ .

Classification

Quantitative

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