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.