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$t$-PPP as the class of total NP-search problems reducible to finding a$t$-collision in a mapping from$(t-1) N + 1$pigeons to$N$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$t$increases, and also give a natural condition on the parameters$t_{1}, t_{2}$that captures exactly when$t_{1}$-PPP and$t_2$-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.