Generalizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak {a}_e$ and $\mathfrak {a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . We also show that there are $\Pi ^1_1$ tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside $\Pi ^1_1$ witnesses for $\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$ .

Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

This systematic literature review aimed to provide an overview of the characteristics and methods used in studies applying the disability-adjusted life years (DALY) concept for infectious diseases within European Union (EU)/European Economic Area (EEA)/European Free Trade Association (EFTA) countries and the United Kingdom. Electronic databases and grey literature were searched for articles reporting the assessment of DALY and its components. We considered studies in which researchers performed DALY calculations using primary epidemiological data input sources. We screened 3053 studies of which 2948 were excluded and 105 studies met our inclusion criteria. Of these studies, 22 were multi-country and 83 were single-country studies, of which 46 were from the Netherlands. Food- and water-borne diseases were the most frequently studied infectious diseases. Between 2015 and 2022, the number of burden of infectious disease studies was 1.6 times higher compared to that published between 2000 and 2014. Almost all studies (97%) estimated DALYs based on the incidence- and pathogen-based approach and without social weighting functions; however, there was less methodological consensus with regards to the disability weights and life tables that were applied. The number of burden of infectious disease studies undertaken across Europe has increased over time. Development and use of guidelines will promote performing burden of infectious disease studies and facilitate comparability of the results.

We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$ . In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$ .

We study higher analogues of the classical independence number on $\omega $ . For $\kappa $ regular uncountable, we denote by $i(\kappa )$ the minimal size of a maximal $\kappa $ -independent family. We establish ZFC relations between $i(\kappa )$ and the standard higher analogues of some of the classical cardinal characteristics, e.g., $\mathfrak {r}(\kappa )\leq \mathfrak {i}(\kappa )$ and $\mathfrak {d}(\kappa )\leq \mathfrak {i}(\kappa )$ . For $\kappa $ measurable, assuming that $2^{\kappa }=\kappa ^{+}$ we construct a maximal $\kappa $ -independent family which remains maximal after the $\kappa $ -support product of $\lambda $ many copies of $\kappa $ -Sacks forcing. Thus, we show the consistency of $\kappa ^{+}=\mathfrak {d}(\kappa )=\mathfrak {i}(\kappa )<2^{\kappa }$ . We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.

Motivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa =\lambda ^+$ for some $\lambda \geq \omega $ and $\mathfrak {b}(\kappa )=\kappa ^+$ then $\mathfrak {a}_e(\kappa )=\mathfrak {a}_p(\kappa )=\kappa ^+$ . If, additionally, $2^{<\lambda }=\lambda $ then $\mathfrak {a}_g(\kappa )=\kappa ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak {d}(\kappa )$ in terms of $\mathfrak {r}(\kappa )$ , including $\mathfrak {d}(\kappa )\leq \mathfrak {r}_\sigma (\kappa )\leq \operatorname {\mathrm {cf}}([\mathfrak {r}(\kappa )]^\omega )$ , and $\mathfrak {d}(\kappa )\leq \mathfrak {r}(\kappa )$ whenever $\mathfrak {r}(\kappa )<\mathfrak {b}(\kappa )^{+\kappa }$ or $\operatorname {\mathrm {cf}}(\mathfrak {r}(\kappa ))\leq \kappa $ holds.

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń’s diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ${\rm{\Delta }}_3^1$ well-order of the reals.

Assuming that every set is constructible, we find a ${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of $\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a ${\text{\Pi }}_1^1 $ maximal cofinitary group in L.

We prove the consistency of

$$~~\text{add}\left( \mathcal{N} \right)<\operatorname{cov}\left( \mathcal{N} \right)<\mathfrak{p}\text{=}\mathfrak{s}\text{=}\mathfrak{g}< \text{add}\left( \mathcal{M} \right)=\text{cof}\left( \mathcal{M} \right)<\mathfrak{a}=\mathfrak{r}=\text{non}\left( N \right)=\mathfrak{c}$$

$\text{ZFC}$

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

We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.