Let [n]=1,…,n, let A⊆ 2^[n], and suppose A satisfies Condition 1: there is a filter F⊆ 2^[n] and a bijection A mapsto F_A from A to F such that A⊆ F_A for every A∈ A and [A,F_A]cap[B,F_B]=emptyset for distinct A,B∈ A, where [A,B]=C:A⊆ C⊆ B. Reimer's condition conjecture. There is an element x∈[n] that belongs to at least half the sets of A. The conjecture asks whether Reimer's sufficient condition alone implies the conclusion of the union-closed sets conjecture. The paper's abstract states that this implication is false, so the conjecture is refuted.

Reimer's condition conjecture

Let $[n]=\{1,\ldots,n\}$ , let ${\mathcal A}\subseteq 2^{[n]}$ , and suppose ${\mathcal A}$ satisfies Condition 1: there is a filter ${\mathcal F}\subseteq 2^{[n]}$ and a bijection $A\mapsto F_A$ from ${\mathcal A}$ to ${\mathcal F}$ such that $A\subseteq F_A$ for every $A\in{\mathcal A}$ and $[A,F_A]\cap[B,F_B]=\emptyset$ for distinct $A,B\in{\mathcal A}$ , where $[A,B]=\{C:A\subseteq C\subseteq B\}$ . Reimer's condition conjecture. There is an element $x\in[n]$ that belongs to at least half the sets of ${\mathcal A}$ . The conjecture asks whether Reimer's sufficient condition alone implies the conclusion of the union-closed sets conjecture. The paper's abstract states that this implication is false, so the conjecture is refuted.

The conjecture, arising from Tim Gowers’s 2016 polymath project, asks whether Reimer’s sufficient condition forces an element to occur in at least half the sets. Raz’s 2017 paper reports a counterexample.

Known results

Raz, 2017: a counterexample on $[8]$ with $11$ sets, where every element occurs in exactly $5$ sets; no element reaches half the family.
Raz, 2017: the counterexample has smallest possible universe size and family size: no example exists with $n<8$ or $|\mathcal A|<11$ .
2024: a later paper reports infinitely many counterexamples satisfying Reimer’s conditions, including examples with prescribed lower bounds on member-set sizes.

Current status (as of September 2026): The exact conjecture is reported refuted by Raz’s counterexample, with infinitely many further examples reported in 2024; independent verification is not recorded in this scan.

Reimer's condition conjecture

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Progress summary

Solutions 0