  We study the complexity of computational problems arising from existence
theorems in extremal combinatorics. For some of these problems, a solution is
guaranteed to exist based on an iterated application of the Pigeonhole
Principle. This results in the definition of a new complexity class within
TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of
PPP, as well as numerous previously unclassified total problems, including
search problems related to Ramsey's theorem, the Sunflower theorem, the
Erd\H{o}s-Ko-Rado lemma, and K\"onig's lemma. Whether the first two of these
four problems are PLC-complete is an important open question which we pursue;
in contrast, we show that the latter two are PPP-complete. Finally, we reframe
PPP as an optimization problem, and define a hierarchy of such problems related
to Tur\'an's theorem.
