This talk will survey some applications of infinitary combinatorics to superficially unrelated questions arising in the study of condensed mathematics, strong homology, and the derived limit functors, showing in the process that they all, combinatorially speaking, drink from the same well. At least as interesting as our ZFC answers to each of these questions are the further questions they open onto; we aim to survey these as well. This work is joint with Chris Lambie-Hanson, and carries a third aim of connecting with some of the material of his seminar talk of the preceding week; no dependency between the two talks, however, is intended.