Résume | The goal of this talk is to explore a general method based on trees of elementary submodels which can be used to highly simplify proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we also present the corresponding technique with 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. |