Math Conjectures

Erdős–Rado Sunflower Conjecture

OPENLandmarkConjectureProposed 1960 · Standard version

Canonical statement

For every integer r≥3r\geq3 there is a constant Cr>0C_r>0 such that, for every k≥1k\geq1, every family of more than CrkC_r^k distinct kk-element sets contains distinct A1,…,ArA_1,\ldots,A_r satisfying
Ai∩Aj=Ai′∩Aj′for all i≠j, i′≠j′. A_i\cap A_j=A_{i'}\cap A_{j'} \quad\text{for all }i\ne j,\ i'\ne j'.
(Such a family is an rr-sunflower.)
View source LaTeX
For every integer \(r\geq3\) there is a constant
\(C_r>0\) such that, for every \(k\geq1\), every family of more than
\(C_r^k\) distinct \(k\)-element sets contains distinct
\(A_1,\ldots,A_r\) satisfying
\[
  A_i\cap A_j=A_{i'}\cap A_{j'}
  \quad\text{for all }i\ne j,\ i'\ne j'.
\]
(Such a family is an \(r\)-sunflower.)
COMB-003:[ErdosRado1960Sunflower][AlweissEtAl2021][Rao2026Sunflowers]
Combinatorics

Notes

An rr-sunflower is a family of rr distinct sets whose pairwise intersections all coincide with a common core. Erdős and Rado proved in 1960 that any family of more than k! (r−1)kk!\,(r-1)^k sets of size kk contains an rr-sunflower, and conjectured that the factorial is an artifact of the proof: for each r≥3r\ge3 there should be a constant CrC_r such that more than CrkC_r^k sets already suffice [ErdosRado1960Sunflower].

For decades the bound improved only in lower-order factors, until Alweiss, Lovett, Wu and Zhang introduced a new method based on spread families that reduced it dramatically [AlweissEtAl2021]. Subsequent refinements brought the bound to the form (Crlog⁡k)k(C_r\log k)^k, the current state of the art; the development is recounted in Rao's survey [Rao2026Sunflowers].

The conjecture demands a base independent of kk, so what remains is precisely to remove the log⁡k\log k factor from the base; even the case r=3r=3 is not settled, and the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.