Proof without Words: The Pigeonhole Principle

A Generalized Resolution Proof Schema and the Pigeonhole Principle

The schematic CERES method is a method of cut elimination for proof schemata, that is a sequence of proofs with a recursive construction. Proof schemata can be thought of as a way to circumvent the addition of an induction rule to the LK-calculus. In this work, we formalize a schematic version of the Infinitary Pigeonhole Principle (IPP), in the LKS-calculus [9], and analyse the extracted claus...

The Pigeonhole Principle

Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects. Proof. Trivial. Example 1.1. Among 13 people there are two who have their birthdays in the same month. Example 1.2. There are n married couples. How many of the 2n people must be selected in order to guarantee that one has selected a married couple? Other principles related to the pigeonhole ...

Proof Complexity of Pigeonhole

Proof Complexity of Pigeonhole Principles

The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m > n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studied combinatorial principle. In this survey we try to summarize what is known about its proof comp...

The Topological Pigeonhole Principle for Ordinals

Given a cardinal κ and a sequence (αi )i∈κ of ordinals, we determine the least ordinal (when one exists) such that the topological partition relation