Dirichlet's Box Principle/Corollary

Corollary to Dirichlet's Box Principle

If a set of $n$ distinct objects is partitioned into $k$ subsets, where $0 < k < n$, then at least one subset must contain at least two elements.

Proof

A direct application of the Dirichlet's Box Principle.

$\blacksquare$

Also known as

Dirichlet's Box Principle, in particular its corollary, is also commonly known as the Pigeonhole Principle or Pigeon-Hole Principle:

Suppose you have $n + 1$ pigeons, but have only $n$ holes for them to stay in.

By the Pigeonhole Principle, at least one of the holes houses $2$ pigeons.

It is also known as Dirichlet's Drawer Principle or Dirichlet's Shelf Principle.

Some sources give it as the Letterbox Principle or Letter-Box Principle.

Some sources call it Dirichlet's Principle, but there is more than one theorem named such.

Some sources give this as the Dirichlet Principle.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.6$: Theorem $9$
• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.18$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): pigeonhole principle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): pigeonhole principle
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Pigeonhole Principle

• Weisstein, Eric W. "Dirichlet's Box Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletsBoxPrinciple.html