Introduction to Combinatorics by Erickson Martin J

Author:Erickson, Martin J.
Language: eng
Format: epub
ISBN: 9781118637586
Publisher: Wiley
Published: 2013-06-10T16:00:00+00:00

EXAMPLE 3.7

Figure 3.3 is the directed graph representation of the partial order

Figure 3.3 A partial order of width 3 and length 3.

on the set X = {1, 2, 3, 4, 5, 6, 7}. The arrows required for reflexivity and transitivity are suppressed in the diagram. The length and width of are both 3. For instance, {1, 3, 4} is a chain of length 3 and {1, 2, 6} is an antichain of size 3.

The length and width of a partial order are related to the size of the underlying set by the following result of R. P. Dilworth.

Dilworth’s lemma. In any partial order on a set of mn + 1 elements, there exists a chain of length m + 1 or an antichain of size n + 1.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.