Improved bounds on the size of the smallest representation of relation algebra \(32_{65}\)

Abstract

In this paper, we shed new light on the spectrum of the relation algebra we call \(A_{n}\), which is obtained by splitting the non-flexible diversity atom of \(6_{7}\) into n symmetric atoms. Precisely, show that the minimum value in \(\text {Spec}(A_{n})\) is at most \(2n^{6 + o(1)}\), which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to \(2n^{2} + 4n + 1\), which is roughly double the trivial bound of \(n^{2} + 2n + 3\). In the process, we obtain stronger results regarding \(\text {Spec}(A_{2}) =\text {Spec}(32_{65})\). Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results.