Forcing theory and combinatorics of the real line

Abstract The main purpose of this dissertation is to apply and develop new forcing techniques obtain models where several cardinal characteristics are pairwise different as well force many (even more, continuum many) values that parametrized by reals. In particular, we look at associated with strong measure zero, Yorioka ideals, localization anti-localization cardinals. thesis introduce the property “ F -linked” subsets posets for a given free filter on natural numbers, define properties $\mu $ - $\theta -Knaster” in way. We show -Knaster preserve types unbounded families maximal almost disjoint families. These kinds led development general technique construct $\textrm {Fr}$ (where Frechet ideal) via matrix iterations ${<}\theta -ultrafilter-linked (restricted some level matrix). latter allows proving consistency results about Cichoń’s diagram (without using large cardinals) prove fact that, each ideal, four it different. Another important application three strongly compact cardinals enough can be separated into 10 values. Later on, was shown Goldstern, Kellner, Mejía, Shelah no needed maximum ( J. Eur. Math. Soc. 24 (2022), no. 11, p. 3951–3967). On other hand, deal certain tree forcings including Sacks forcing, these increase covering zero ideal $\mathcal {SN}$ . As consequence, model, such number equal size continuum, which indicates consistently larger than any classical continuum. Even used $\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cov}}(\mathcal {\mathrm{cof}}(\mathcal {SN})$ , first result more two To another direction, provide bounds generalizes Yorioka’s characterization Symbolic Logic 67.4 (2002), 1373–1384). get {\mathrm{add}}(\mathcal {SN})=\operatorname ZFC (via iteration construction). conclude combining creature approaches Kellner Arch. 51.1–2 (2012), 49–70) Fischer, 56.7–8 (2017), 1045–1103) under CH, there proper $\omega ^\omega -bounding poset $\aleph _2$ -cc forces characteristics, reals, one following six types: uniformity numbers ideals both cardinals, respectively. This answers open questions from Klausner Mejía 61 pp. 653–683). prepared Miguel Antonio Cardona-Montoya E-mail : [email protected] URL https://repositum.tuwien.at/handle/20.500.12708/19629