The Erdős–Rado Sunflower Conjecture

Definitions and mathematical statement

An $r$-petal sunflower is a collection of distinct sets $S_1,\ldots,S_r$ with one fixed core $C$ satisfying $S_i\cap S_j=C$ for every $i\ne j$. Equivalently, their parts outside the core are pairwise disjoint; the core may be empty. The conjecture is \[ \forall r\ge3\ \exists C_r>0\ \forall k\ge1:\quad |\mathcal F|>C_r^k,\quad|S|\le k\ (S\in\mathcal F) \Rightarrow\mathcal F\text{ contains an }r\text{-sunflower}. \] The ground set is unrestricted and the exponential base may depend on $r$ but not on $k$.

Formulation and concept references: [S1].

Short English statement

Must every sufficiently large bounded-size set family contain a sunflower, with a threshold only exponential in the set size?

Sources

• S1 Ashwin Rao, The story of sunflowers, Journal of the London Mathematical Society, 2026. https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.70380. Formulation source.