Relation algebras and \(t\)-vertex condition graphs
A notion of superscheme was suggested by \textit{J. D. H. Smith} in [Eur. J. Comb. 15, No.~3, 285-291 (1994; Zbl 0796.05097)] as a relational analogue of a transitive multiplicity-free permutation group acting on a finite set \(Q\). This notion is defined with the aid of four axioms which postulate properties of a family of partitions \(\Gamma^n\) of the direct powers \(Q^{n+2}\) for each natural number \(n\). Some slightening of these axioms leads to the notion of a height \(t\) presuperscheme. Each strongly regular graph \(G=(V,E)\) defines naturally an association scheme with two classes on the set \(V\). The author describes also how to associate to \(G\) a certain presuperscheme. Main result of the paper is Theorem 4.4: If a presuperscheme associated with a strongly regular graph \(G\) is of height \(t\), then the graph \(G\) satisfies the \(t+3\)-vertex condition (in the sense of Zbl 0709.05040 and Zbl 0795.05073). A sketch of a justification of this theorem is presented, which unfortunately does not contain enough details in order to explain the source of a few formulas, appearing as an essential part of the proof.
scientific article; zbMATH DE number 1248172