On Pigeonhole Principles and Ramsey in TFNP

We show that the TFNP problem Ramsey is not black-box reducible to Pigeon, refuting a conjecture of Goldberg and Papadimitriou in the black-box setting. We prove this by giving reductions to Ramsey from a new family of TFNP problems that correspond to generalized versions of the pigeonhole principle, and then proving that these generalized versions cannot be reduced to Pigeon. Formally, we define <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex>-PPP as the class of total NP-search problems reducible to finding a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex>-collision in a mapping from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(t-1) N + 1$</tex> pigeons to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> holes. These classes are closely related to multi-collision resistant hash functions in cryptography. We show that the generalized pigeonhole classes form a hierarchy as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> increases, and also give a natural condition on the parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{1}, t_{2}$</tex> that captures exactly when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{1}$</tex>-PPP and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_2$</tex>-PPP collapse in the black-box setting. Finally, we prove other inclusion and separation results between these generalized Pigeon problems and other previously studied TFNP subclasses, such as PLS, PPA, and PLC. Our separation results rely on new lower bounds in propositional proof complexity based on pseudoexpectation operators, which may be of independent interest.