notes-091119 - MATH 681 Notes Combinatorics and Graph...

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Unformatted text preview: MATH 681 Notes Combinatorics and Graph Theory I 1 Chains and Antichains 1.1 Maximality and maximum-, uh, -ness? To review, the definitions of a chain and antichain: Definition 1. A chain is a totally ordered subset of a poset S ; an antichain is a subset of a poset S in which any two distinct elements are incomparable. Now, we have two distinct concepts of a chain/antichain being as large as possible. One of those concepts is not easily extendable that is to say, a chain or antichain which cant have elements added to make it a larger chain/antichain: Definition 2. A maximal chain (antichain) is one that is not a proper subset of another chain (antichain). Alternatively, a chain C is maximal in a poset ( S, ) if no element of S- C is comparable to every element of C ; an antichain A is maximal if no element of S- A is incomparable to every element of A . Our other definition of largeness is that of simply being of the greatest size in a poset. Definition 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 is known as a posets width . Note, trivially, that a maximum chain/antichain must be maximal: if C is maximum, than there are no chains of size greater than | C | ; if C were a proper subset of a chain D , then | D | > | C | would contradict the above assertion. As an example, let us consider the poset ( { 1 , 2 , 3 , 4 , 5 , 6 , 9 , 12 , 18 } , | ), recalling that a | b is a is a divisor of b . Then we may note that { 1 , 2 , 4 , 12 } is a maximum chain there are no chains of length greater than 4. The maximum chain is not unique: { 1 , 3 , 6 , 18 } is also a maximum chain, as is { 1 , 2 , 6 , 12 } and several others. An example of a maximal chain which is not maximum is { 1 , 5 } : this has only two elements, and thus is not maximum, but no more elements of S can be added to it, since 5 is incomparable with everything except 1 and 5. An example of a nonmaximal chain would be { 1 , 3 , 18 } , because it is a subset of a larger chain (in fact, two larger chains, since either 6 or 9 could be included). Looking at antichains, the largest we could find would have size 4: { 4 , 6 , 9 , 5 } would be an example, and is in fact the unique maximum antichain (uniqueness is not guaranteed; its just how this example works out). We could find several maximal antichains which are not maximum, such as { 2 , 3 , 5 } or { 12 , 18 } , or even { 1 } (which would be maximal because everything is comparable to 1!). And of course, nonmaximal antichains can be found too, such as { 12 , 5 } , which is a subset of the larger antichain { 12 , 9 , 5 } ....
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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notes-091119 - MATH 681 Notes Combinatorics and Graph...

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