Filter-linkedness and its effect on preservation of cardinal characteristics

Preservation of splitting families and cardinal characteristics of the continuum

“…Similar preservation techniques have appeared in different contexts. For instance, concerning preservation of mad (maximal almost disjoint) families, Kunen [Kun80] constructed, under CH, a mad family that can be preserved by Cohen posets; afterwards, Steprāns [Ste93] showed that, after adding ω 1many Cohen reals, there is a mad family of size ℵ 1 that can be preserved in further Cohen extensions; Fischer and Brendle [BF11] constructed a Hechlertype poset H A with support (any uncountable set) A that adds a mad family indexed by A, which can be preserved not only in further Cohen extensions but after other concrete FS iterations, thus generalizing Steprāns' result because H ω1 = C ω1 ; [FFMM18,Mej19a] showed that any such mad family added by H A can be preserved by some general type of FS iterations, but the most general result so far was shown in [BCM21]: Any κ-Fr-Knaster poset preserves κ-strong-Md-families (with κ uncountable regular; the mad family added by H κ is of such type).…”

“…In this direction [GKS19] constructed a forcing model, using four strongly compact cardinals, where all the ten (non-dependent) values of Cichoń's diagram are pairwise different (a situation we call Cichoń's Maximum), as in Figure 3(A). This was improved later in [BCM21] by only using three strongly compact cardinals; finally in [GKMSa] it was shown that no large cardinals are needed for Cichoń's Maximum.…”

The covering number of the strong measure zero ideal can be above almost everything else

“…The following idea may be useful to answer the question above. Quite recently, the first and second authors with Brendle [5] constructed a ccc poset forcing add(N ) = add(M) < cov(N ) = non(M) < cov(M) = non(N ) < cof(M) = cof(N ).…”

Cichoń’s diagram and localisation cardinals

“…6 When referring to a single pair of functions in a congenial sequence, we will call this a congenial pair of functions. 5 As a matter of fact, in the inductive definitions of these sequences, we will only be demanding that they be far larger than some other term, and we define them in some appropriate way to ensure this property; making them even larger would not pose any problems. 6 Property (iii) here corresponds to the assumption in [13,Theorem 3.1], but is more specific.…”

“…We turn our attention towards related work and open questions. Several recent results [5,[7][8][9][10][11]15,16] have constructed models in which eight or even all ten conceivably different cardinal characteristics in Cichoń's diagram take different values. The constructions involved are all finite support iterations, however, which necessarily means the left side of Cichoń's diagram must be less than or equal to the right side, in particular non(M) ≤ cov(M) (since the cofinality of the iteration length lies between these two cardinal characteristics).…”