Papers › Comer Schemes, Relation Algebras, and the Flexible Atom Conjecture
Comer Schemes, Relation Algebras, and the Flexible Atom Conjecture
Jeremy F. Alm, David A. Andrews, Michael Levet
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In this paper, we consider relational structures arising from Comer's finite field construction, where the cosets need not be sum free. These Comer schemes generalize the notion of a Ramsey scheme and may be of independent interest. As an application, we give the first finite representation of 34₆₅. This leaves 33₆₅ as the only remaining relation algebra in the family N₆₅ with a flexible atom that is not known to be finitely representable. Motivated by this, we complement our upper bounds with some lower bounds. Using a SAT solver, we show that 33₆₅ is not finitely representable on fewer than $24$ points, and that 33₆₅ does not admit a cyclic group representation on fewer than $120$ points. We also employ a SAT solver to show that 34₆₅ is not representable on fewer than $24$ points.
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