Abstract 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.

Abstract We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.