Flexible Atom Conjecture for finite integral relation algebras
Let be a finite integral relation algebra (RA), and suppose that has a flexible atom. A relation algebra is representable over a finite cyclic group if it embeds into a Comer relation algebra associated with such a group. Flexible Atom Conjecture. Any finite integral RA with a flexible atom embeds in some Comer RA, hence is representable over a finite cyclic group. The conjecture proposes that the observed ability to realize suitable forbidden schemes in Comer relation algebras extends to every finite integral relation algebra with a flexible atom. Establishing it would provide a route toward understanding the apparent quasirandom behavior of Comer relation algebras.
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The flexible atom conjecture for finite integral relation algebras
Let a finite integral relation algebra be an algebra whose Boolean reduct has finitely many elements and whose identity is an atom. An atom is flexible when it does not participate in any forbidden diversity cycles.
Flexible atom conjecture. Every finite integral relation algebra with a flexible atom is representable over a finite set.
The conjecture concerns finite representations of relation algebras and would rule out finite integral relation algebras with flexible atoms that are representable only over infinite sets. The supplied passage does not indicate whether the conjecture is currently open or resolved.
source: Jeremy F. Alm and Michael Levet, “Directed Ramsey and Anti-Ramsey Schemes and the Flexible Atom Conjecture”, arXiv:1901.06781 (2023).
References
Primary source
Jeremy F. Alm, David A. Andrews and Michael Levet, “Comer Schemes, Relation Algebras, and the Flexible Atom Conjecture”, arXiv:1905.11914 (2025).
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