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.
翻译:组合器中的许多古典理论在足够庞大的收藏对象中确立了子结构的出现。 众所周知的例子有 Ramsey 单色子谱和 Erd\H{o}s- Rado 太阳花列。 相应的全部搜索问题的隐含版本已知为 PWPP-hard ; 这里的“ implici” 表示该收藏由多尺寸电路代表, 引出大量天体。 我们显示, 来自极端组合器的另外几个众所周知的词结构结构结构。 我们的工作确定了一些来自极端组合组合的太阳学证据, 包括Erd\H{ o}s- KO- Rado、 Sperner 和 Cayley 的公式 。 这给 PWPP 和 PPP 带来了完整的问题。 这与 Ramsey 和 Erd\ H{H}- Rado 的问题不同, 因为这些问题在 PWPP 中仍然难以被包含。 除了大幅扩展 PWPP 和 PPPPP 之外, 我们的工作还确定了一些来自 Excistrubal cotor 校 证据的关键特性, 证明 。 存在一些关键的太阳证据, 证据证据的特性证据证明, 其存在证明, 使得 无法将使得这些精确的精确的系统进行精确的系统化工作成为这些系统化的系统化。