Ramsey quantifiers are a natural object of study not only for logic and computer science but also for the formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.

Ilmenite concentrates obtained from heavy mineral deposits in the Murray Basin, southeastern Australia contain appreciable quantities (up to 10 wt.%) of an unusual hydrated Fe titanate mineral in the form of orange to red coloured translucent grains. The grains are weakly magnetic and have a low apparent density (<3.3 g cm–3) due to microprosity, which enables them to be easily concentrated in a pure state. Samples from different deposits have similar analyses of 67–69% TiO2, 8 –9% Fe2O3, 2–3% FeO, 11 –12% H2O and 8 –9% of intra-pore impurities, mainly SiO2 and Al2O3. The powder pattern for the mineral closely matches that for pseudorutile; having a hexagonal subcell with a = 2.844 (1) Å, c = 4.575 (1) Å. The idealized composition range of the Fe titanate, calculated from analyses and structure refinements, is [FeTi6O12(OH)3].3H2O to [FeTi6O11(OH)5].2H2O. The molecular water is nonstructural, probably adsorbed on the surfaces of nano-scale domains of the mineral. Electron microprobe and scanning electron microscopy studies provide evidence for formation of the mineral by replacement reactions acting on MgO-rich ferrian ilmenite grains. The mineral represents a distinct intermediate in the chemical weathering of ilmenite, having a pseudorutile-like structure but a leucoxene-like composition. It is unusual in not having undergone any recrystallization to phases such as rutile and anatase.

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley’s lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.