PPP-Completeness and Extremal Combinatorics

Abstract: Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erd\H{o}s-Rado sunflower lemma. Implicit versions of the corresponding total search problems are known to be PWPP-hard; here "implici" means that the collection is represented by a poly-sized circuit inducing an exponentially large number of objects. We show that several other well-known theorems from extremal combinatorics - including Erd\H{o}s-Ko-Rado, Sperner, and Cayley's formula - give rise to complete problems for PWPP and PPP. This is in contrast to the Ramsey and Erd\H{o}s-Rado problems, for which establishing inclusion in PWPP has remained elusive. Besides significantly expanding the set of problems that are complete for PWPP and PPP, our work identifies some key properties of combinatorial proofs of existence that can give rise to completeness for these classes. Our completeness results rely on efficient encodings for which finding collisions allows extracting the desired substructure. These encodings are made possible by the tightness of the bounds for the problems at hand (tighter than what is known for Ramsey's theorem and the sunflower lemma). Previous techniques for proving bounds in TFNP invariably made use of structured algorithms. Such algorithms are not known to exist for the theorems considered in this work, as their proofs "from the book" are non-constructive.