Abstract We study whether it is possible to apply twisted Laplace spectra (and, if so, how many are necessary) deduce isometry of some well-known examples isospectral manifolds in the literature, due Schüth (simply connected manifolds), Ikeda (lens spaces), Vignéras/Linowitz and Voight (arithmetic surfaces), Milnor (lattice examples), Doyle Rossetti (Tetra Didi), Sunada (based on group-theoreti...

We present several examples of hereditary classes finite structures satisfying the joint embedding property and weak amalgamation property, but failing cofinal property. These include a continuum-sized family undirected graphs, as well an example due to Pouzet with countably categorical generic limit.