Splitting families of sets in ZFC

Small Sets in Convex Geometry and Formal Independence over Zfc

To each closed subset S of a finite dimensional Euclidean space corresponds a σ-ideal of sets J (S) which is σ-generated over S by the convex subsets of S. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals to other types of ideals, such as the ideals w...

Degrees of M-fuzzy families of independent L-fuzzy sets

The present paper studies fuzzy matroids in view of degree. First wegeneralize the notion of $(L,M)$-fuzzy independent structure byintroducing the degree of $M$-fuzzy family of independent $L$-fuzzysets with respect to a mapping from $L^X$ to $M$. Such kind ofdegrees is proved to satisfy some axioms similar to those satisfiedby $(L,M)$-fuzzy independent structure. ...

Analytic countably splitting families

Mad families, splitting families and large continuum

Let κ < λ be regular uncountable cardinals. Using a finite support iteration of ccc posets we obtain the consistency of b = a = κ < s = λ. If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

Splitting NP-Complete Sets

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al., complete sets for all of the following complexity classes are m-mitotic: NP, coNP, ⊕P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the c...