The pigeonhole bootstrap

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 ...

The Pigeonhole Bootstrap1 by Art

Recently there has been much interest in data that, in statistical language, may be described as having a large crossed and severely unbalanced random effects structure. Such data sets arise for recommender engines and information retrieval problems. Many large bipartite weighted graphs have this structure too. We would like to assess the stability of algorithms fit to such data. Even for linea...

Proof Complexity of Pigeonhole

Goldbach ’ s Pigeonhole

Edward Early ([email protected]) received his Ph.D. from the Massachusetts Institute of Technology. He is an associate professor of mathematics at St. Edward’s University in Austin, TX. His research is mostly in combinatorics, but he enjoys dabbling in number theory. He also likes spending time with his family and training in martial arts. Patrick Kim ([email protected]) is a senior at S...

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...