We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

We show that intersection graphs of compact convex sets in R n bounded aspect ratio have asymptotic dimension at most 2 + 1 . More generally, we this is the case for systems subsets any metric space Assouad–Nagata satisfy following condition: For each r , s > 0 and every point p number pairwise-disjoint elements diameter least system are distance from by a function /

Cichoń’s diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with ideals of null sets, meager countable and $\sigma$-compact subsets irrationals. It is consistent that all entries are pairwise different (apart from $\operatorname{add}\mathcal{M}$ $\operatorname{cof}\mathcal{M}$, which provably equal to other entries). However, consistency proofs...

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth ergodic diffeomorphisms certain compact manifolds are unclassifiable up to measure-isomorphism. In this paper we show the uniform circular systems used work Foreman—Weiss admit real-analytic realizations on two-dimensional torus. As consequence obtain same anti-classification result for another application exis...

The problem of bipartite ranking, where instances are labeled positive or negative and the goal is to learn a scoring function that minimizes the probability of mis-ranking a pair of positive and negative instances (or equivalently, that maximizes the area under the ROC curve), has been widely studied in recent years. A dominant theoretical and algorithmic framework for the problem has been to ...

We describe a new approach based on tropical optimization techniques to solve the problem of rating alternatives from pairwise comparison data. The problem is formulated to approximate, in the logChebyshev sense, pairwise comparison matrices by reciprocal matrices of unit rank, and then represented in general terms of tropical mathematics as a tropical optimization problem. The optimization pro...

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with א1-many. Even taking the compact space to be the unit interval, this already implies the failure of the Con...

Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpubl...

We construct two models containing exactly one supercompact cardinal in which all nonsupercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness ...