Erdős–Rado Sunflower Conjecture

Resolve the Erdős–Rado sunflower conjecture by proving or refuting that there exists a constant C depending only on the number of petals w such that any family F of sets of size at most k that contains no sunflower with w petals has cardinality at most C^k.

Background

A sunflower with w petals is a collection of sets whose pairwise intersections are all equal (the core). Erdős and Rado showed that any sufficiently large family of k-element sets contains a sunflower, with a classical bound of k!·(w−1)^k. They further posited a much stronger quantitative statement: that forbidding a w-petal sunflower suffices to limit the family to C(w)^k sets, independent of k.

Recent progress established a robust sunflower lemma that implies any family of size O((w log k)^k) contains a w-petal sunflower, narrowing the gap with the conjecture to a logarithmic factor in k. The extra log k factor is unavoidable for robust sunflowers, but it may be removable for standard sunflowers. Despite advances, the original Erdős–Rado conjecture remains unresolved.