Narrow Results

In this paper, we improve Galvin’s Theorem for ultrafilters which are $p$-point limits of $p$-points. This implies that in all the canonical inner models up to a superstrong cardinal, every $\kappa$-complete ultrafilter over a measurable cardinal $\kappa$ satisfies the Galvin property. On the other hand, we prove that supercompact cardinals always carry non-Galvin $\kappa$-complete ultrafilters. Finally, we prove that $♢(\kappa )$ implies the existence of a $\kappa$-complete filter which extends the club filter and fails to satisfy the Galvin property. This answers questions [8, Question 5.22], [4, Question 3.4] and questions, [7, Question 4.5], [6, Question 2.26].

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $\kappa$, if $\kappa$ is weakly compact, then $♢(\kappa )$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [O. B. -Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19(1) (2019) 1950002].

There are mines of elemental carbon such as graphite. It is the most stable form of elemental carbon at 25°C, under 1 atm (Section 3.7). A second form of pure or nearly pure elemental carbon is represented by diamonds (Section 3.12). Other forms of elemental carbon are produced by combustion or heat treatment of wood (Section 3.6), bio-polymers such as paper, cotton (cellulose), or synthetic polymers such as viscose (Section 11.10.1) and polyacrylonitrile (Section 8.2.7). Nanoparticles such as graphene (Section 3.7), fullerenes (Section 3.8), nanotubes (Section 3.9), and quantum dots of carbon (Section 3.10) are available that find biomedical applications and are used in the manufacture of nanometric objects useful for electronics, optoelectronics, photophysics, energy and the environment protection. Nanoparticles must be handled with care as they can be toxic…