In joint work with Macpherson, Steinhorn and Wolf on multidimensional asymptotic classes, we introduced ring-measurable structures. In these, every definable set is assigned a "size" in an ordered ring, subject to additivity and Fubini-type conditions on definable families. This generalises both the measurable structures of Macpherson and Steinhorn and the asymptotic counting in pseudofinite structures. We asked there whether ring-measurability implies simplicity. In ongoing work with Macpherson and Steinhorn, we show that the generic triangle-free graph (the Henson graph H_3), which is not simple, is not ring-measurable: counting configurations in two different ways yields polynomial identities with no admissible common solution. I will explain the framework and sketch the argument. I will then discuss the generic bowtie-free graph, which defines copies of H_3 that are not fully embedded, so that the argument does not transfer directly.