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The Pigeon Hole Principle

A function f:X\longrightarrow Y is said to be 11 (read one-to-one) when f(x)\neq f(x') for all x,x'\in X with x\neq x'. A 11 function is also called an injection or we say that f is injective.

The Pigeon Hole Principle

A function \(f:X\longrightarrow Y\) is said to be \(1\)\(1\) (read one-to-one) when \(f(x)\neq f(x')\) for all \(x,x'\in X\) with \(x\neq x'\). A \(1\)\(1\) function is also called an injection or we say that \(f\) is injective. When \(f:X\longrightarrow Y\) is \(1\)\(1\), we note that \(|X|\le |Y|\). Conversely, we have the following self-evident statement, which is popularly called the Pigeon Hole principle.

In more casual language, if you must put \(n+1\) pigeons into \(n\) holes, then you must put two pigeons into the same hole.

Here is a classic result, whose proof follows immediately from the .

Theorem: Erdős/Szekeres Theorem

If \(m\) and \(n\) are non-negative integers, then any sequence of \(mn+1\) distinct real numbers either has an increasing subsequence of \(m+1\) terms, or it has a decreasing subsequence of \(n+1\) terms.

Proof

Let \(\sigma=(x_1,x_2,x_3,\dots,x_{mn+1})\) be a sequence of \(mn+1\) distinct real numbers. For each \(i=1,2,\dots,mn+1\), let \(a_i\) be the maximum number of terms in a increasing subsequence of \(\sigma\) with \(x_i\) the first term. Also, let \(b_i\) be the maximum number of terms in a decreasing subsequence of \(\sigma\) with \(x_i\) the last term. If there is some \(i\) for which \(a_i\ge m+1\), then \(\sigma\) has an increasing subsequence of \(m+1\) terms. Conversely, if for some \(i\), we have \(b_i\ge n+1\), then we conclude that \(\sigma\) has a decreasing subsequence of \(n+1\) terms.

It remains to consider the case where \(a_i\le m\) and \(b_i\le n\) for all \(i=1,2,\dots,mn+1\). Since there are \(mn\) ordered pairs of the form \((a,b)\) where \(1\le a\le m\) and \(1\le b\le n\), we conclude from the Pigeon Hole principle that there must be integers \(i_1\) and \(i_2\) with \(1\le i_1\lt i_2\le mn+1\) for which \((a_{i_1},b_{i_1})=(a_{i_2},b_{i_2})\). Since \(x_{i_1}\) and \(x_{i_2}\) are distinct, we either have \(x_{i_1}\lt x_{i_2}\) or \(x_{i_1}>x_{i_2}\). In the first case, any increasing subsequence with \(x_{i_2}\) as its first term can be extended by prepending \(x_{i_1}\) at the start. This shows that \(a_{i_1}>a_{i_2}\). In the second case, any decreasing sequence of with \(x_{i_1}\) as its last element can be extended by adding \(x_{i_2}\) at the very end. This shows \(b_{i_2}>b_{i_1}\).

In , we will explore some powerful generalizations of the . All these results have the flavor of the general assertion that total disarray is impossible.

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Àwọn Àtòjọ-ẹ̀yàn

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Àwọn ààyè ojú-ìwé yìí tí a fi pamọ́ láti Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Tí a fi pamọ́ síì níbẹ̀; àwọn àwọn àṣiṣe ní wa.

Diẹ̀ nínú Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Recurrences and generating functions Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton Colouring and planar graphs