Chain antichain theorem

2020-01-17 14:36

Robert Dilworths 1950 theorem may be restated thus: if a poset has ab 1 elements then it has a chain of length a 1 or an antichain of length b 1. In this form it generalises a classic 1935 result ofDe nition 3. A maximum or longest chain (largest antichain) is one which is of the greatest size possible. The size of the longest chain is known as a posets height. The size of the largest antichain chain antichain theorem

Aug 05, 2018 Chain and antichain are two very important concepts of posets and hase diagram.

Chain antichain theorem free

A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition.

Antichain. A chain in S is a subset C of S in which each pair of elements is comparable; that is, C is totally ordered. An antichain in S is a subset A of S in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in A.

It just says you can't find an antichain and a partition into chains, such that every chain intersects the antichain in exactly one element. Konig's Theorem stated that in bipartite graphs the maximum matching has the same size as the minimum vertex cover.

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Dilworths Theorem. Theorem (1950) A poset of width w can be partitioned into w chains. Proofs of Dilworths Theorem. Fulkerson (1954) Used bipartite matching algorithm (network flows) to find minimum chain partition and maximum antichain simultaneously.

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