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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Characterizes the height of any finite partially ordered set</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, in the areas of <a href="/wiki/Order_theory" title="Order theory">order theory</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, <b>Mirsky's theorem</b> characterizes the height of any finite <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> in terms of a partition of the order into a minimum number of <a href="/wiki/Antichain" title="Antichain">antichains</a>. It is named for <a href="/wiki/Leon_Mirsky" title="Leon Mirsky">Leon Mirsky</a> (<a href="#CITEREFMirsky1971">1971</a>) and is closely related to <a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a> on the widths of partial orders, to the <a href="/wiki/Perfect_graph" title="Perfect graph">perfection</a> of <a href="/wiki/Comparability_graph" title="Comparability graph">comparability graphs</a>, to the <a href="/wiki/Gallai%E2%80%93Hasse%E2%80%93Roy%E2%80%93Vitaver_theorem" title="Gallai–Hasse–Roy–Vitaver theorem">Gallai–Hasse–Roy–Vitaver theorem</a> relating <a href="/wiki/Longest_path_problem" title="Longest path problem">longest paths</a> and <a href="/wiki/Graph_coloring" title="Graph coloring">colorings</a> in graphs, and to the <a href="/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem" title="Erdős–Szekeres theorem">Erdős–Szekeres theorem</a> on monotonic subsequences. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_theorem">The theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=1" title="Edit section: The theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The height of a partially ordered set is defined to be the maximum cardinality of a <a href="/wiki/Antichain" title="Antichain">chain</a>, a <a href="/wiki/Total_order" title="Total order">totally ordered</a> subset of the given partial order. For instance, in the set of positive integers from 1 to <i>N</i>, ordered by <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>, one of the largest chains consists of the <a href="/wiki/Power_of_two" title="Power of two">powers of two</a> that lie within that range, from which it follows that the height of this partial order is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\lfloor \log _{2}N\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\lfloor \log _{2}N\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5134781cf9623910a437fe362e93fecaf78f4d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.544ex; height:2.843ex;" alt="{\displaystyle 1+\lfloor \log _{2}N\rfloor }"></span>. </p><p>Mirsky's theorem states that, for every finite 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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\lfloor \log _{2}N\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\lfloor \log _{2}N\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5134781cf9623910a437fe362e93fecaf78f4d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.544ex; height:2.843ex;" alt="{\displaystyle 1+\lfloor \log _{2}N\rfloor }"></span> 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. </p><p>To prove the existence of a partition into a small number of antichains for an arbitrary finite partially ordered set, consider for every element <i>x</i> the chains that have <i>x</i> as their largest element, and let <i>N</i>(<i>x</i>) denote the size of the largest of these <i>x</i>-maximal chains. Then each set <i>N</i><sup>−1</sup>(<i>i</i>), consisting of elements that have equal values of <i>N</i>, 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. </p> <div class="mw-heading mw-heading2"><h2 id="Related_results">Related results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=2" title="Edit section: Related results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Dilworth's_theorem"><span id="Dilworth.27s_theorem"></span>Dilworth's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=3" title="Edit section: Dilworth's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mirsky was inspired by <a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a>, 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 <a href="/wiki/Order_dimension" title="Order dimension">order dimension</a> two, the two theorems coincide (a chain in the <a href="/wiki/Majorization" title="Majorization">majorization</a> 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. </p><p>Mirsky's theorem and Dilworth's theorem are also related to each other through the theory of <a href="/wiki/Perfect_graph" title="Perfect graph">perfect graphs</a>. An undirected graph is <a href="/wiki/Perfect_graph" title="Perfect graph">perfect</a> if, in every <a href="/wiki/Induced_subgraph" title="Induced subgraph">induced subgraph</a>, the <a href="/wiki/Chromatic_number" class="mw-redirect" title="Chromatic number">chromatic number</a> equals the size of the largest clique. In the <a href="/wiki/Comparability_graph" title="Comparability graph">comparability graph</a> 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 <a href="/wiki/Complement_graph" title="Complement graph">complement graph</a> of a comparability graph is perfect. The <a href="/wiki/Perfect_graph_theorem" title="Perfect graph theorem">perfect graph theorem</a> of <a href="#CITEREFLovász1972">Lovász (1972)</a> 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 (<a href="#CITEREFGolumbic1980">Golumbic 1980</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Gallai–Hasse–Roy–Vitaver_theorem"><span id="Gallai.E2.80.93Hasse.E2.80.93Roy.E2.80.93Vitaver_theorem"></span>Gallai–Hasse–Roy–Vitaver theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=4" title="Edit section: Gallai–Hasse–Roy–Vitaver theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mirsky's theorem can be restated in terms of <a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graphs</a> (representing a partially ordered set by <a href="/wiki/Reachability" title="Reachability">reachability</a> of their vertices), as the statement that there exists a <a href="/wiki/Graph_homomorphism" title="Graph homomorphism">graph homomorphism</a> from a given directed acyclic graph <i>G</i> to a <i>k</i>-vertex <a href="/wiki/Tournament_(graph_theory)" title="Tournament (graph theory)">transitive tournament</a> if and only if there does not exist a homomorphism from a (<i>k</i> + 1)-vertex <a href="/wiki/Path_graph" title="Path graph">path graph</a> to <i>G</i>. For, the largest path graph that has a homomorphism to <i>G</i> 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 <i>G</i> is not acyclic, and is a form of the <a href="/wiki/Gallai%E2%80%93Hasse%E2%80%93Roy%E2%80%93Vitaver_theorem" title="Gallai–Hasse–Roy–Vitaver theorem">Gallai–Hasse–Roy–Vitaver theorem</a> on <a href="/wiki/Graph_coloring" title="Graph coloring">graph colorings</a> and <a href="/wiki/Orientation_(graph_theory)" title="Orientation (graph theory)">orientations</a> (<a href="#CITEREFNešetřilOssona_de_Mendez2012">Nešetřil & Ossona de Mendez 2012</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Erdős–Szekeres_theorem"><span id="Erd.C5.91s.E2.80.93Szekeres_theorem"></span>Erdős–Szekeres theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=5" title="Edit section: Erdős–Szekeres theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It follows from either Dilworth's theorem or Mirsky's theorem that, in every partially ordered set of <i>rs</i> + 1 elements, there must exist a chain of <i>r</i> + 1 elements or an antichain of <i>s</i> + 1 elements. <a href="#CITEREFMirsky1971">Mirsky (1971)</a> uses this observation, applied to a partial order of order dimension two, to prove the <a href="/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem" title="Erdős–Szekeres theorem">Erdős–Szekeres theorem</a> that in every sequence of <i>rs</i> + 1 totally ordered elements there must exist a monotonically increasing subsequence of <i>r</i> + 1 elements or a monotonically decreasing subsequence of <i>s</i> + 1 elements. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=6" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> κ, there exist partially ordered sets that have no infinite chain and that do not have an antichain partition with κ or fewer antichains (<a href="#CITEREFSchmerl2002">Schmerl 2002</a>). </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mirsky%27s_theorem&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDilworth1950" class="citation cs2"><a href="/wiki/Robert_P._Dilworth" title="Robert P. Dilworth">Dilworth, Robert P.</a> (1950), "A Decomposition Theorem for Partially Ordered Sets", <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, <b>51</b> (1): 161–166, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969503">10.2307/1969503</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969503">1969503</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=A+Decomposition+Theorem+for+Partially+Ordered+Sets&rft.volume=51&rft.issue=1&rft.pages=161-166&rft.date=1950&rft_id=info%3Adoi%2F10.2307%2F1969503&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969503%23id-name%3DJSTOR&rft.aulast=Dilworth&rft.aufirst=Robert+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMirsky%27s+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolumbic1980" class="citation cs2"><a href="/wiki/Martin_Charles_Golumbic" title="Martin Charles Golumbic">Golumbic, Martin Charles</a> (1980), "5.7. Coloring and other problems on comparability graphs", <i>Algorithmic Graph Theory and Perfect Graphs</i>, New York: Academic Press, pp. 132–135, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-289260-7" title="Special:BookSources/0-12-289260-7"><bdi>0-12-289260-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0562306">0562306</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.7.+Coloring+and+other+problems+on+comparability+graphs&rft.btitle=Algorithmic+Graph+Theory+and+Perfect+Graphs&rft.place=New+York&rft.pages=132-135&rft.pub=Academic+Press&rft.date=1980&rft.isbn=0-12-289260-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D562306%23id-name%3DMR&rft.aulast=Golumbic&rft.aufirst=Martin+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMirsky%27s+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLovász1972" class="citation cs2"><a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">Lovász, László</a> (1972), "Normal hypergraphs and the perfect graph conjecture", <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>, <b>2</b> (3): 253–267, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2872%2990006-4">10.1016/0012-365X(72)90006-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Normal+hypergraphs+and+the+perfect+graph+conjecture&rft.volume=2&rft.issue=3&rft.pages=253-267&rft.date=1972&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2872%2990006-4&rft.aulast=Lov%C3%A1sz&rft.aufirst=L%C3%A1szl%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMirsky%27s+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMirsky1971" class="citation cs2"><a href="/wiki/Leon_Mirsky" title="Leon Mirsky">Mirsky, Leon</a> (1971), "A dual of Dilworth's decomposition theorem", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>78</b> (8): 876–877, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2316481">10.2307/2316481</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2316481">2316481</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+dual+of+Dilworth%27s+decomposition+theorem&rft.volume=78&rft.issue=8&rft.pages=876-877&rft.date=1971&rft_id=info%3Adoi%2F10.2307%2F2316481&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2316481%23id-name%3DJSTOR&rft.aulast=Mirsky&rft.aufirst=Leon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMirsky%27s+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNešetřilOssona_de_Mendez2012" class="citation cs2"><a href="/wiki/Jaroslav_Ne%C5%A1et%C5%99il" title="Jaroslav Nešetřil">Nešetřil, Jaroslav</a>; <a href="/wiki/Patrice_Ossona_de_Mendez" title="Patrice Ossona de Mendez">Ossona de Mendez, Patrice</a> (2012), "Theorem 3.13", <i>Sparsity: Graphs, Structures, and Algorithms</i>, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 42, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-27875-4">10.1007/978-3-642-27875-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-27874-7" title="Special:BookSources/978-3-642-27874-7"><bdi>978-3-642-27874-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2920058">2920058</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theorem+3.13&rft.btitle=Sparsity%3A+Graphs%2C+Structures%2C+and+Algorithms&rft.place=Heidelberg&rft.series=Algorithms+and+Combinatorics&rft.pages=42&rft.pub=Springer&rft.date=2012&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2920058%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-27875-4&rft.isbn=978-3-642-27874-7&rft.aulast=Ne%C5%A1et%C5%99il&rft.aufirst=Jaroslav&rft.au=Ossona+de+Mendez%2C+Patrice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMirsky%27s+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmerl2002" class="citation cs2">Schmerl, James H. (2002), "Obstacles to extending Mirsky's theorem", <i><a href="/wiki/Order_(journal)" title="Order (journal)">Order</a></i>, <b>19</b> (2): 209–211, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1016541101728">10.1023/A:1016541101728</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1922918">1922918</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:26514679">26514679</a></cite><span 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href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a class="mw-selflink selflink">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mirsky%27s_theorem&oldid=1184448688">https://en.wikipedia.org/w/index.php?title=Mirsky%27s_theorem&oldid=1184448688</a>"</div></div> <div 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