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In [[mathematics]], in the areas of [[order theory]] and [[combinatorics]], '''Mirsky's theorem''' characterizes the height of any finite [[partially ordered set]] in terms of a partition of the order into a minimum number of [[antichain]]s. It is named for {{harvs|first=Leon|last=Mirsky|authorlink=Leon Mirsky|year=1971|txt}} and is closely related to [[Dilworth's theorem]] on the widths of partial orders, to the [[perfect graph|perfection]] of [[comparability graph]]s, to the [[Gallai–Hasse–Roy–Vitaver theorem]] relating [[longest path problem|longest paths]] and [[graph coloring|colorings]] in graphs, and to the [[Erdős–Szekeres theorem]] on monotonic subsequences.
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==The theorem==
The height of a partially ordered set is defined to be the maximum cardinality of a [[antichain|chain]], a [[total order|totally ordered]] subset of the given partial order. For instance, in the set of positive integers from 1 to ''N'', ordered by [[divisibility]], one of the largest chains consists of the [[power of two|powers of two]] that lie within that range, from which it follows that the height of this partial order is <math>1+\lfloor\log_2 N\rfloor</math>.
 
Mirsky's theorem states that, for every partially ordered set, the height also equals the minimum number of antichains (subsets in which no pair of elements are ordered) into which the set may be partitioned. In such a partition, every two elements of the longest chain must go into two different antichains, so the number of antichains is always greater than or equal to the height; another formulation of Mirsky's theorem is that there always exists a partition for which the number of antichains equals the height. Again, in the example of positive integers ordered by divisibility, the numbers can be partitioned into the antichains {1}, {2,3}, {4,5,6,7}, etc. There are <math>1+\lfloor\log_2 N\rfloor</math> sets in this partition, and within each of these sets, every pair of numbers forms a ratio less than two, so no two numbers within one of these sets can be divisible.
 
To prove the existence of a partition into a small number of antichains for an arbitrary finite partially ordered set, consider for every element ''x'' the chains that have ''x'' as their largest element, and let ''N''(''x'') denote the size of the largest of these ''x''-maximal chains. Then each set ''N''<sup>−1</sup>(''i''), consisting of elements that have equal values of ''N'', is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain. In his original proof, Mirsky constructs the same partition inductively, by choosing an antichain of the maximal elements of longest chains, and showing that the length of the longest chain among the remaining elements is reduced by one.
 
==Relation to Dilworth's theorem==
Mirsky was inspired by [[Dilworth's theorem]], stating that, for every partially ordered set, the maximum size of an antichain equals the minimum number of chains in a partition of the set into chains. For sets of [[order dimension]] two, the two theorems coincide (a chain in the [[majorization]] ordering of points in general position in the plane is an antichain in the set of points formed by a 90° rotation from the original set, and vice versa) but for more general partial orders the two theorems differ, and (as Mirsky observes) Dilworth's theorem is more difficult to prove.
 
Mirsky's theorem and Dilworth's theorem are also related to each other through the theory of [[perfect graph]]s. An undirected graph is [[Perfect graph|perfect]] if, in every [[induced subgraph]], the [[chromatic number]] equals the size of the largest clique. In the [[comparability graph]] of a partially ordered set, a clique represents a chain and a coloring represents a partition into antichains, and induced subgraphs of comparability graphs are themselves comparability graphs, so Mirsky's theorem states that comparability graphs are perfect. Analogously, Dilworth's theorem states that every [[complement graph]] of a comparability graph is perfect. The [[perfect graph theorem]] of {{harvtxt|Lovász|1972}} states that the complements of perfect graphs are always perfect, and can be used to deduce Dilworth's theorem from Mirsky's theorem and vice versa {{harv|Golumbic|1980}}.
 
==Relation to the Gallai–Hasse–Roy–Vitaver theorem==
Mirsky's theorem can be restated in terms of [[directed acyclic graph]]s (representing a partially ordered set by [[reachability]] of their vertices), as the statement that there exists a [[graph homomorphism]] from a given directed acyclic graph ''G'' to a ''k''-vertex [[Tournament (graph theory)|transitive tournament]] if and only if there does not exist a homomorphism from a (''k''&nbsp;+&nbsp;1)-vertex [[path graph]] to ''G''. For, the largest path graph that has a homomorphism to ''G'' gives the longest chain in the reachability ordering, and the sets of vertices with the same image in a homomorphism to a transitive tournament form a partition into antichains. This statement generalizes to the case that ''G'' is not acyclic, and is a form of the [[Gallai–Hasse–Roy–Vitaver theorem]] on [[graph coloring]]s and [[Orientation (graph theory)|orientations]] {{harv|Nešetřil|Ossona de Mendez|2012}}.
 
==Relation to the Erdős–Szekeres theorem==
It follows from either Dilworth's theorem or Mirsky's theorem that, in every partially ordered set of ''rs''&nbsp;+&nbsp;1 elements, there must exist either a chain of ''r''&nbsp;+&nbsp;1 elements or an antichain of ''s''&nbsp;+&nbsp;1 elements. {{harvtxt|Mirsky|1971}} uses this observation, applied to a partial order of order dimension two, to prove the [[Erdős–Szekeres theorem]] that in every sequence of ''rs''&nbsp;+&nbsp;1 totally ordered elements there must exist either a monotonically increasing subsequence of ''r''&nbsp;+&nbsp;1 elements or a monotonically decreasing subsequence of ''s''&nbsp;+&nbsp;1 elements.
 
==Extensions==
Mirsky's theorem extends immediately to infinite partially ordered sets with finite height. However, the relation between the length of a chain and the number of antichains in a partition into antichains does not extend to infinite cardinalities: for every infinite [[cardinal number]] &kappa;, there exist partially ordered sets that have no infinite chain and that do not have an antichain partition with &kappa; or fewer antichains {{harv|Schmerl|2002}}.
 
==References==
*{{citation
| last = Dilworth | first = Robert P.
| authorlink = Robert P. Dilworth
| title = A Decomposition Theorem for Partially Ordered Sets
| jstor = 1969503
| journal = [[Annals of Mathematics]]
| volume = 51
| issue = 1 | pages = 161–166 | year = 1950
| doi = 10.2307/1969503}}.
*{{citation
| last = Golumbic | first = Martin Charles | authorlink = Martin Charles Golumbic
| contribution = 5.7. Coloring and other problems on comparability graphs
| isbn = 0-12-289260-7
| location = New York
| mr = 562306
| pages = 132–135
| publisher = Academic Press
| title = Algorithmic Graph Theory and Perfect Graphs
| year = 1980}}.
*{{citation
| last = Lovász | first = László | authorlink = László Lovász
| year = 1972
| title = Normal hypergraphs and the perfect graph conjecture
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| volume = 2
| issue = 3
| pages = 253–267
| doi = 10.1016/0012-365X(72)90006-4}}.
*{{citation
| authorlink = Leon Mirsky
| last = Mirsky | first = Leon
| title = A dual of Dilworth's decomposition theorem
| jstor = 2316481
| journal = [[American Mathematical Monthly]]
| volume = 78
| issue = 8 | year = 1971 | pages = 876–877
| doi = 10.2307/2316481}}.
*{{citation
| last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil
| last2 = Ossona de Mendez | first2 = Patrice
| contribution = Theorem 3.13
| doi = 10.1007/978-3-642-27875-4
| isbn = 978-3-642-27874-7
| location = Heidelberg
| mr = 2920058
| page = 42
| publisher = Springer
| series = Algorithms and Combinatorics
| title = Sparsity: Graphs, Structures, and Algorithms
| volume = 28
| year = 2012}}.
*{{citation
| last = Schmerl | first = James H.
| doi = 10.1023/A:1016541101728
| issue = 2
| journal = [[Order (journal)|Order]]
| mr = 1922918
| pages = 209–211
| title = Obstacles to extending Mirsky's theorem
| volume = 19
| year = 2002}}.
 
[[Category:Order theory]]
[[Category:Theorems in combinatorics]]
[[Category:Articles containing proofs]]
[[Category:Perfect graphs]]

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As you realize in an excellent world there are correct ways to do issues and there are different methods to 'pores and skin a cat' as the outdated saying goes. Cutlers both previous and current day are conscious of many ways to fulfill an finish and this is but one method to adjust a liner lock. It is usually Leatherman Pocket Knife Review attainable for a talented cutler to adjust the cease pin diameter to achieve the identical outcomes somewhat than peen the lock. In some cases the place a knife by no means seen earlier than is opened up after being disassembled you sometimes find that the lock has already been adjusted by peening.

Designing quality tools for the good outdoor, Finest Made Co. revamps some of the cherished European pocket knives with the release of the Mercator K55K Locking Knife. The Mercator K55K was first produced in the course of the 19th century by Heinrich Kaufmann & Söhne, in Solingen, Germany and is still offered in Germany to today. Although it nonetheless options its hallmark cat silhouette and conventional slim design structure, this rendition is nothing like your grandfather’s piece as it sees a smoother opening, a powder-coated steel handle, as well as a much finer carbon metal C75 locking blade. This Mercator mannequin is presently available on-line for $forty two USD.

Tool Logic gives you knowledgeable-grade and functional folding survival knife made of top quality metal building. The SL-Pro 2 comes with a sturdy Zytel frame and a liner-fashion lock that lets you open and use the knife with a single-handed operation. Moreover, the blade measures three inches in length, and it's extremely sharp for accuracy in reducing a wide range of things. For a cheap value of $27, the knife has the whole lot you will need for a survival gear. It is also in a position to maintain a superb edge longer, so you will not be disappointed upon purchasing this outstanding tool to your wilderness journey.

The Recon 1 is a tactically designed knife that has a AUS-8A Teflon coated blade to stop glare, reduce friction, and look robust. The blade has a clip point which makes it great for stabbing. AUS-8A is a high quality Japanese steel that is a lot better than different steels on this price vary. The 4 the best pocket knife reviews.Com inch lengthy blade provides you loads of room for slicing, dicing, and all different chopping wants. The knife has a thumb stub that opens the blade with ease. The stud is adjustable. a design that enables for single hand opening. This is usually passive via a thumb stud, flipper or hole on the blade.

The Chilly Steel Hold Out folders look lots like little folding daggers. The uniform shape of each the handle and blade remind me of historical stone knives that I've seen in museums. And this is a very practical shape, by the way. Chilly Steel’s replace on a timeless knife model how to sharpen a pocket knife permits you the safe grip you need in a knife, coupled with simple opening and the strong locking of Chilly Metal’s Tri-Advert Lock mechanism. This knife is light weight at 2.5 ounces, however Japanese AUS 8A stainless-steel and G-10 handles give it plenty of power. It has a three inch blade.