  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.
