maths.free › Combinatorics & Graph Theory › Partially Ordered Sets › Dilworth's Chain Covering Theorem and its Dual

Dilworth's Chain Covering Theorem and its Dual

In this section, we prove the following theorem of R.P. Dilworth, which is truly one of the classic results of combinatorial mathematics.

Dilworth's Chain Covering Theorem and its Dual

In this section, we prove the following theorem of R.P. Dilworth, which is truly one of the classic results of combinatorial mathematics.

Before proceeding with the proof of Dilworth's theorem later in this section, we pause to discuss the dual version for partitions into antichains, as it is even easier to prove.

Theorem: Dual of Dilworth's Theorem

If \(\PXP\) is a poset and \(\height(P)=h\), then there exists a partition \(X=A_1\cup A_2\cup\dots\cup A_h\), where \(A_i\) is an antichain for \(i=1,2,\dots,h\). Furthermore, there is no partition using fewer antichains.

Proof

For each \(x\in X\), let \(\height(x)\) be the largest integer \(t\) for which there exists a chain \[\begin{aligned}\end{aligned}\] with \(x=x_t\). Evidently, \(\height(x)\le h\) for all \(x\in X\). Then for each \(i=1,2,\dots,h\), let \(A_i=\{x\in X:\height(x)=i\}\). It is easy to see that each \(A_i\) is an antichain, as if \(x,y\in A_i\) are such that \(x\lt y\), then there is a chain \(x_1\lt x_2\lt \cdots\lt x_i=x \lt x_{i+i}=y\), so \(\height(y)\ge i+1\). Since \(\height(P)=h\), there is a maximum chain \(C=\{x_1,x_2,\dots,x_h\}\). If it were possible to partition \(\bfP\) into \(t\lt h\) antichains, then by the Pigeon Hole Principle, one of the antichains would contain two points from \(C\), but this is not possible.

When \(\PXP\) is a poset, a point \(x\in X\) with \(\height(x)=1\) is called a minimal point of \(\bfP\). We denote the set of all minimal points of a poset \(\PXP\) by \(\min(X,P)\).Since we use the notation \(\bfP= (X,P)\) for a poset, the set of minimal elements can be denoted by \(\min(\bfP)\) or \(\min(X,P)\). This convention will be used for all set valued and integer valued functions of posets.

The argument given for the proof of yields an efficient algorithm, one that is defined recursively. Set \(\bfP_0= \bfP\). If \(\bfP_i\) has been defined and \(\bfP_i\neq \emptyset\), let \(A_i=\min(\bfP_i)\) and then let \(\bfP_{i+1}\) denote the subposet remaining when \(A_i\) is removed from \(\bfP_i\).

In , we illustrate the antichain partition provided by this algorithm for the \(17\) point poset from . The darkened points form a chain of size\(5\).

Maximal antichains and maximum antichains are defined analogously.

Condensed: the full section is in Keller & Trotter, Applied Combinatorics.

이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.

연습 (1)

먼저 종이에 각각을 시도해 보세요. 확인하기 위해 답을 드러내세요. 확인된 답은 각 단계의 솔버에서 열 수 있습니다.

Consider implementing an algorithm from on the poset of . First, verify that you agree with the labeling of the points with their heights, which assigns the points to antichains. Explain why this poset is a good illustration of why finding the chain of five points is best done by starting with a point labeled \(5\) and working down instead of starting with a point labeled \(1\) and working up.

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

사람들이 묻는 질문

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.

이 페이지의 일부는 다음에서 변경되었습니다. Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). 여기서 압축하고 다시 설명; 오류는 우리의.

에 더 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