1941:
2320:
However, the theorem does not extend so simply to partially ordered sets in which the width, and not just the cardinality of the set, is infinite. In this case the size of the largest antichain and the minimum number of chains needed to cover the partial order may be very different from each other.
62:
A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of the smallest chain decomposition are again equal.
80:
chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain decomposition is its number of chains. The width of the
2644:
is defined as the minimum number of chains needed to define the antimatroid, and
Dilworth's theorem can be used to show that it equals the width of an associated partial order; this connection leads to a polynomial time algorithm for convex dimension.
75:
in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into
2582:
2352:
A dual of
Dilworth's theorem states that the size of the largest chain in a partial order (if finite) equals the smallest number of antichains into which the order may be partitioned. This is called
1356:
2439:
of any perfect graph is also perfect. Therefore, the complement of any comparability graph is perfect; this is essentially just
Dilworth's theorem itself, restated in graph-theoretic terms (
701:
1930:
881:
443:
284:
1695:
1266:
1520:
1140:
1027:
927:
2190:
both of which have the same size. But the only nontrivial chains in the partial order are pairs of elements corresponding to the edges in the graph, so the nontrivial chains in
350:
1645:
1060:
531:
487:
1572:
1455:
1409:
1727:
1216:
960:
2907:
630:
1383:
987:
795:
748:
585:
558:
377:
2427:
equals the size of the largest clique. Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms. By the
1859:
1753:
1546:
1481:
2306:
2999:
2596:
1833:
1813:
1793:
1773:
1612:
1592:
1429:
1286:
1183:
1160:
1100:
1080:
835:
815:
768:
721:
605:
397:
304:
241:
218:
198:
178:
158:
138:
107:
1944:
Proof of
Dilworth's theorem via KĹ‘nig's theorem: constructing a bipartite graph from a partial order, and partitioning into chains according to a matching
3986:
3265:
2502:
3969:
3499:
3335:
3139:
2781:
3243:
3816:
1949:
2416:
of a comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements.
2388:, is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain.
2404:
formed from a partial order by creating a vertex per element of the order, and an edge connecting any two comparable elements. Thus, a
4029:
3952:
3811:
3035:
3806:
3442:
3524:
3058:
2409:
2630:
3843:
3763:
3162:
3086:
2941:
1291:
3437:
3628:
3557:
2825:
3531:
3519:
3482:
3457:
3432:
3386:
3355:
3462:
3452:
635:
2285:
colors; every color class in a proper coloring of the incomparability graph must be a chain. By the assumption that
1864:
840:
402:
3828:
3328:
1957:
3801:
3467:
2443:). Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.
3733:
3360:
3119:
17:
3274:
2083:
contains vertices corresponding to the same element on both sides of the bipartition) and no two elements of
4024:
3981:
3964:
3217:
3115:
2233:
Dilworth's theorem for infinite partially ordered sets states that a partially ordered set has finite width
2123:
chains. Therefore, we have constructed an antichain and a partition into chains with the same cardinality.
1650:
1221:
3893:
3509:
2480:
1486:
1105:
992:
886:
246:
3053:
2633:
on monotone subsequences can be interpreted as an application of
Dilworth's theorem to partial orders of
4019:
3871:
3706:
3697:
3566:
3401:
3365:
3321:
2795:
2428:
2405:
309:
40:
1617:
1032:
3959:
3918:
3908:
3898:
3643:
3606:
3596:
3576:
3561:
3030:, Advances in Mathematics (Springer), vol. 7, New York: Springer, Theorem 5.6, p. 60,
2769:
2484:
48:
492:
448:
3886:
3797:
3743:
3702:
3692:
3581:
3514:
3477:
2397:
2353:
2347:
2270:
3253:
3225:, IMA Volumes in Mathematics and its Applications, vol. 72, Springer-Verlag, pp. 111–131
1551:
1434:
1388:
3925:
3778:
3687:
3677:
3618:
3536:
3286:
3189:
3103:
2958:
2924:
2890:
2847:
2812:
2790:
1700:
1188:
932:
56:
3998:
3838:
3472:
3261:
3935:
3913:
3773:
3758:
3738:
3541:
3296:
3201:
3135:
3031:
2777:
2205:
This connection to bipartite matching allows the width of any partial order to be computed in
3025:
3748:
3601:
3171:
3127:
3095:
3067:
3010:
2950:
2916:
2874:
2839:
2804:
2436:
2424:
2413:
2401:
3185:
3149:
3045:
2970:
2886:
1361:
965:
773:
726:
563:
536:
355:
3930:
3713:
3591:
3586:
3571:
3396:
3381:
3181:
3145:
3041:
2966:
2902:
2882:
2865:
2830:
2634:
2452:
2322:
2206:
1953:
3487:
3237:
2878:
2356:. Its proof is much simpler than the proof of Dilworth's theorem itself: for any element
1838:
1732:
1525:
1460:
1940:
610:
3848:
3833:
3823:
3682:
3660:
3638:
3209:
2984:
2266:
1818:
1798:
1778:
1758:
1597:
1577:
1414:
1271:
1168:
1145:
1085:
1065:
820:
800:
753:
706:
590:
382:
289:
226:
203:
183:
163:
143:
123:
92:
2861:"Recognition algorithms for orders of small width and graphs of small Dilworth number"
81:
partial order is defined as the common size of the antichain and chain decomposition.
4013:
3947:
3903:
3881:
3753:
3623:
3611:
3416:
3193:
3072:
3015:
2978:
2851:
2420:
1948:
Like a number of other results in combinatorics, Dilworth's theorem is equivalent to
32:
3299:
3288:
Recognition
Algorithms for Orders of Small Width and Graphs of Small Dilworth Number
2860:
2599:
also concerns antichains in a power set and can be used to prove
Sperner's theorem.
3768:
3650:
3633:
3551:
3391:
3344:
3213:
3205:
3157:
2894:
2765:
2603:
2326:
77:
28:
2828:(1988), "Combinatorial representation and convex dimension of convex geometries",
2317:-colorable incomparability graph, and thus has the desired partition into chains.
3204:(1995), "Variations on the monotone subsequence theme of Erdős and Szekeres", in
3974:
3667:
3546:
3411:
3126:, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 42,
3081:
2936:
2641:
2577:{\displaystyle \operatorname {width} (B_{n})={n \choose \lfloor {n/2}\rfloor }.}
110:
24:
2905:(1956), "Note on Dilworth's decomposition theorem for partially ordered sets",
3942:
3876:
3717:
3248:
3131:
3993:
3866:
3672:
3304:
2464:
72:
44:
3267:
Lecture Notes in
Combinatorics and Probability, Lecture 10: Perfect Graphs
89:
The following proof by induction on the size of the partially ordered set
3788:
3655:
3406:
2281:
as vertices, with an edge between every two incomparable elements) using
2776:, Annals of Discrete Mathematics, vol. 21, Elsevier, p. viii,
2622:
2. Therefore, by
Dilworth's theorem, the width of this partial order is
2126:
To prove KĹ‘nig's theorem from
Dilworth's theorem, for a bipartite graph
3176:
3107:
2962:
2928:
2843:
2816:
2293:, and by the finite version of Dilworth's theorem, every finite subset
16:"Chain decomposition" redirects here. For the path decomposition, see
3099:
2954:
2920:
2808:
2338:
discusses analogues of Dilworth's theorem in the infinite setting.
2039:, such that each edge in the graph contains at least one vertex in
55:
of the partially order. The theorem is named for the mathematician
1939:
140:
be a finite partially ordered set. The theorem holds trivially if
3317:
3313:
3056:(1972), "Normal hypergraphs and the perfect graph conjecture",
2793:(1950), "A Decomposition Theorem for Partially Ordered Sets",
2939:(1994), "A proof of Dilworth's chain decomposition theorem",
2587:
In other words, a largest family of incomparable subsets of
2859:
Felsner, Stefan; Raghavan, Vijay; Spinrad, Jeremy (2003),
2412:
in a comparability graph corresponds to an antichain. Any
1971:
elements, using KĹ‘nig's theorem, define a bipartite graph
2981:; Kleitman, Daniel J. (1976), "The structure of Sperner
2408:
in a comparability graph corresponds to a chain, and an
51:
needed to cover all elements. This number is called the
2606:, the subinterval forms an antichain with cardinality
3160:(1963), "On Dilworth's theorem in the infinite case",
3084:(1971), "A dual of Dilworth's decomposition theorem",
2696:
2694:
2685:
2325:Îş there is an infinite partially ordered set of width
1956:
matching and several other related theorems including
47:
of incomparable elements equals the minimum number of
2987:
2505:
2332:
whose partition into the fewest chains has Îş chains.
2241:
chains. For, suppose that an infinite partial order
1867:
1841:
1821:
1801:
1781:
1761:
1735:
1703:
1653:
1620:
1600:
1580:
1554:
1528:
1489:
1463:
1437:
1417:
1391:
1364:
1294:
1274:
1224:
1191:
1171:
1148:
1108:
1088:
1068:
1035:
995:
968:
935:
889:
843:
823:
803:
776:
756:
729:
709:
638:
613:
593:
566:
539:
495:
451:
405:
385:
358:
312:
292:
249:
229:
206:
186:
166:
146:
126:
95:
3285:
Felsner, S.; Raghavan, V. & Spinrad, J. (1999),
3238:
Equivalence of seven major theorems in combinatorics
2384:), consisting of elements that have equal values of
2305:-colorable incomparability graph. Therefore, by the
1351:{\displaystyle \{a\}\cup \{z\in C_{i}:z\leq x_{i}\}}
3859:
3787:
3726:
3496:
3425:
3374:
2182:. By Dilworth's theorem, there exists an antichain
2993:
2576:
1924:
1853:
1827:
1807:
1787:
1767:
1747:
1721:
1689:
1639:
1606:
1586:
1566:
1540:
1514:
1475:
1449:
1423:
1403:
1377:
1350:
1280:
1260:
1210:
1177:
1154:
1134:
1094:
1074:
1054:
1021:
981:
954:
921:
875:
829:
809:
789:
762:
742:
715:
695:
624:
599:
579:
552:
525:
481:
437:
391:
371:
344:
298:
278:
235:
212:
192:
172:
152:
132:
101:
2565:
2534:
2249:, meaning that there are at most a finite number
2908:Proceedings of the American Mathematical Society
2614:chains is easy to achieve: for each odd integer
2194:form a matching in the graph. The complement of
1963:To prove Dilworth's theorem for a partial order
2023:. By KĹ‘nig's theorem, there exists a matching
696:{\displaystyle A:=\{x_{1},x_{2},\dots ,x_{k}\}}
1925:{\displaystyle \{a\},C_{1},C_{2},\dots ,C_{k}}
876:{\displaystyle A_{i}\cap C_{j}\neq \emptyset }
438:{\displaystyle A_{0}\cap C_{i}\neq \emptyset }
223:By induction, we assume that for some integer
3329:
2618:in , form a chain of the numbers of the form
2602:If we order the integers in the interval by
2342:Dual of Dilworth's theorem (Mirsky's theorem)
2253:of elements in any antichain. For any subset
2099:in the same chain whenever there is an edge (
8:
3124:Sparsity: Graphs, Structures, and Algorithms
2724:
2559:
2543:
2440:
2265:chains (if it exists) may be described as a
2229:Extension to infinite partially ordered sets
2202:with the same cardinality as this matching.
1874:
1868:
1716:
1710:
1684:
1660:
1509:
1496:
1345:
1307:
1301:
1295:
1255:
1231:
690:
645:
273:
267:
2748:
2142:), form a partial order on the vertices of
3987:Positive cone of a partially ordered group
3336:
3322:
3314:
2372:) denote the size of the largest of these
2341:
2237:if and only if it may be partitioned into
2091:be a family of chains formed by including
3175:
3071:
3014:
3003:Journal of Combinatorial Theory, Series A
2986:
2673:
2610:. A partition of this partial order into
2564:
2550:
2546:
2533:
2531:
2519:
2504:
1916:
1897:
1884:
1866:
1840:
1820:
1800:
1780:
1760:
1734:
1702:
1652:
1631:
1619:
1599:
1579:
1553:
1527:
1503:
1488:
1462:
1436:
1416:
1390:
1369:
1363:
1339:
1320:
1293:
1273:
1223:
1202:
1190:
1170:
1147:
1126:
1113:
1107:
1087:
1067:
1040:
1034:
1013:
1000:
994:
973:
967:
946:
934:
913:
900:
888:
861:
848:
842:
822:
802:
781:
775:
755:
734:
728:
708:
684:
665:
652:
637:
612:
592:
571:
565:
544:
538:
494:
450:
423:
410:
404:
384:
363:
357:
336:
317:
311:
291:
248:
228:
205:
185:
165:
145:
125:
94:
3970:Positive cone of an ordered vector space
2700:
2661:
2591:is obtained by selecting the subsets of
2063:that do not correspond to any vertex in
2654:
1614:disjoint chains, as required. Next, if
1558:
1493:
1441:
1395:
264:
2736:
2712:
2432:
2335:
114:
1690:{\displaystyle i\in \{1,2,\dots ,k\}}
1261:{\displaystyle i\in \{1,2,\dots ,k\}}
587:that belongs to an antichain of size
7:
2686:Felsner, Raghavan & Spinrad 2003
2597:Lubell–Yamamoto–Meshalkin inequality
1515:{\displaystyle A\setminus \{x_{i}\}}
1135:{\displaystyle x_{j}\not \geq x_{i}}
1022:{\displaystyle x_{i}\not \geq x_{j}}
922:{\displaystyle y\in A_{i}\cap C_{j}}
279:{\displaystyle P':=P\setminus \{a\}}
3219:Discrete Probability and Algorithms
2487:states that a maximum antichain of
2423:if, in every induced subgraph, the
1411:does not have an antichain of size
3497:Properties & Types (
2879:10.1023/B:ORDE.0000034609.99940.fb
2538:
2392:Perfection of comparability graphs
2364:as their largest element, and let
2321:In particular, for every infinite
2277:(a graph that has the elements of
2087:are comparable to each other. Let
870:
432:
345:{\displaystyle C_{1},\dots ,C_{k}}
180:has at least one element, and let
14:
3953:Positive cone of an ordered field
2942:The American Mathematical Monthly
2213:-element partial orders of width
797:. Fix arbitrary distinct indices
3807:Ordered topological vector space
2360:, consider the chains that have
1640:{\displaystyle a\not \geq x_{i}}
1062:. By interchanging the roles of
1055:{\displaystyle x_{i}\not \geq y}
2447:Width of special partial orders
2376:-maximal chains. Then each set
352:and has at least one antichain
2525:
2512:
2170:, and there exists an edge in
1431:. Induction then implies that
1102:in this argument we also have
526:{\displaystyle i=1,2,\dots ,k}
482:{\displaystyle i=1,2,\dots ,k}
1:
3764:Series-parallel partial order
3163:Israel Journal of Mathematics
3087:American Mathematical Monthly
2640:The "convex dimension" of an
3443:Cantor's isomorphism theorem
3244:"Dual of Dilworth's Theorem"
3073:10.1016/0012-365X(72)90006-4
3016:10.1016/0097-3165(76)90077-7
2186:and a partition into chains
1567:{\displaystyle P\setminus K}
1450:{\displaystyle P\setminus K}
1404:{\displaystyle P\setminus K}
59:, who published it in 1950.
3483:Szpilrajn extension theorem
3458:Hausdorff maximal principle
3433:Boolean prime ideal theorem
2595:that have median size. The
2079:elements (possibly more if
1722:{\displaystyle A\cup \{a\}}
1211:{\displaystyle a\geq x_{i}}
955:{\displaystyle y\leq x_{j}}
39:states that, in any finite
4046:
3829:Topological vector lattice
2483:or, notationally, (2, ⊆).
2345:
2217:can be recognized in time
2059:be the set of elements of
2051:have the same cardinality
560:be the maximal element in
243:the partially ordered set
160:is empty. So, assume that
15:
4030:Theorems in combinatorics
3351:
3132:10.1007/978-3-642-27875-4
3120:Ossona de Mendez, Patrice
3024:Harzheim, Egbert (2005),
1936:Proof via KĹ‘nig's theorem
43:, the maximum size of an
3438:Cantor–Bernstein theorem
3122:(2012), "Theorem 3.13",
2774:Topics on Perfect Graphs
2725:Berge & Chvátal 1984
2441:Berge & Chvátal 1984
2198:forms a vertex cover in
2031:, and a set of vertices
1932:, completing the proof.
1729:is an antichain of size
1522:is an antichain of size
1358:. Then by the choice of
750:be an antichain of size
200:be a maximal element of
18:Heavy path decomposition
3982:Partially ordered group
3802:Specialization preorder
2749:Edelman & Saks 1988
2475:—essentially {1, 2, …,
2419:An undirected graph is
2307:De Bruijn–Erdős theorem
2261:, a decomposition into
1958:Hall's marriage theorem
1935:
962:, by the definition of
3468:Kruskal's tree theorem
3463:Knaster–Tarski theorem
3453:Dushnik–Miller theorem
3216:; et al. (eds.),
2995:
2631:Erdős–Szekeres theorem
2578:
1945:
1926:
1855:
1835:can be covered by the
1829:
1809:
1789:
1769:
1749:
1723:
1691:
1641:
1608:
1588:
1568:
1542:
1516:
1483:disjoint chains since
1477:
1451:
1425:
1405:
1379:
1352:
1282:
1262:
1212:
1179:
1156:
1136:
1096:
1076:
1056:
1023:
983:
956:
923:
877:
831:
811:
791:
764:
744:
723:is an antichain. Let
717:
697:
626:
601:
581:
554:
527:
483:
439:
393:
373:
346:
300:
280:
237:
214:
194:
174:
154:
134:
103:
2996:
2873:(4): 351–364 (2004),
2796:Annals of Mathematics
2579:
2429:perfect graph theorem
2271:incomparability graph
1943:
1927:
1856:
1830:
1810:
1790:
1770:
1750:
1724:
1692:
1642:
1609:
1589:
1569:
1543:
1517:
1478:
1452:
1426:
1406:
1380:
1378:{\displaystyle x_{i}}
1353:
1283:
1263:
1213:
1185:. Suppose first that
1180:
1157:
1142:. This verifies that
1137:
1097:
1077:
1057:
1024:
984:
982:{\displaystyle x_{j}}
957:
924:
878:
832:
812:
792:
790:{\displaystyle x_{i}}
765:
745:
743:{\displaystyle A_{i}}
718:
698:
627:
602:
582:
580:{\displaystyle C_{i}}
555:
553:{\displaystyle x_{i}}
528:
484:
440:
394:
374:
372:{\displaystyle A_{0}}
347:
301:
281:
238:
215:
195:
175:
155:
135:
104:
41:partially ordered set
3960:Ordered vector space
3059:Discrete Mathematics
2985:
2503:
1865:
1839:
1819:
1799:
1779:
1759:
1733:
1701:
1651:
1618:
1598:
1578:
1552:
1526:
1487:
1461:
1435:
1415:
1389:
1362:
1292:
1272:
1222:
1189:
1169:
1146:
1106:
1086:
1066:
1033:
993:
989:. This implies that
966:
933:
887:
841:
821:
801:
774:
754:
727:
707:
636:
611:
591:
564:
537:
493:
449:
403:
383:
356:
310:
290:
247:
227:
204:
184:
164:
144:
124:
109:is based on that of
93:
3798:Alexandrov topology
3744:Lexicographic order
3703:Well-quasi-ordering
2791:Dilworth, Robert P.
2398:comparability graph
1854:{\displaystyle k+1}
1748:{\displaystyle k+1}
1541:{\displaystyle k-1}
1476:{\displaystyle k-1}
3779:Transitive closure
3739:Converse/Transpose
3448:Dilworth's theorem
3300:"Dilworth's Lemma"
3297:Weisstein, Eric W.
3202:Steele, J. Michael
3177:10.1007/BF02759806
3116:Nešetřil, Jaroslav
2991:
2844:10.1007/BF00143895
2824:Edelman, Paul H.;
2574:
2313:itself also has a
2209:. More precisely,
1946:
1922:
1851:
1825:
1805:
1785:
1765:
1745:
1719:
1687:
1637:
1604:
1594:can be covered by
1584:
1564:
1538:
1512:
1473:
1457:can be covered by
1447:
1421:
1401:
1375:
1348:
1278:
1258:
1208:
1175:
1152:
1132:
1092:
1072:
1052:
1019:
979:
952:
919:
873:
827:
807:
787:
760:
740:
713:
693:
625:{\displaystyle P'}
622:
597:
577:
550:
523:
479:
435:
389:
369:
342:
296:
286:can be covered by
276:
233:
210:
190:
170:
150:
130:
99:
57:Robert P. Dilworth
37:Dilworth's theorem
27:, in the areas of
4007:
4006:
3965:Partially ordered
3774:Symmetric closure
3759:Reflexive closure
3502:
3141:978-3-642-27874-7
2994:{\displaystyle k}
2783:978-0-444-86587-8
2563:
2496:has size at most
2485:Sperner's theorem
1828:{\displaystyle P}
1808:{\displaystyle P}
1788:{\displaystyle a}
1768:{\displaystyle P}
1607:{\displaystyle k}
1587:{\displaystyle P}
1424:{\displaystyle k}
1281:{\displaystyle K}
1178:{\displaystyle P}
1165:We now return to
1162:is an antichain.
1155:{\displaystyle A}
1095:{\displaystyle j}
1075:{\displaystyle i}
830:{\displaystyle j}
810:{\displaystyle i}
763:{\displaystyle k}
716:{\displaystyle A}
703:. We claim that
600:{\displaystyle k}
392:{\displaystyle k}
299:{\displaystyle k}
236:{\displaystyle k}
213:{\displaystyle P}
193:{\displaystyle a}
173:{\displaystyle P}
153:{\displaystyle P}
133:{\displaystyle P}
102:{\displaystyle P}
4037:
3749:Linear extension
3498:
3478:Mirsky's theorem
3338:
3331:
3324:
3315:
3310:
3309:
3291:
3281:
3279:
3273:, archived from
3272:
3257:
3252:, archived from
3226:
3224:
3196:
3179:
3158:Perles, Micha A.
3152:
3110:
3076:
3075:
3048:
3019:
3018:
3000:
2998:
2997:
2992:
2973:
2931:
2903:Fulkerson, D. R.
2897:
2854:
2826:Saks, Michael E.
2819:
2786:
2752:
2746:
2740:
2734:
2728:
2722:
2716:
2710:
2704:
2698:
2689:
2683:
2677:
2671:
2665:
2659:
2583:
2581:
2580:
2575:
2570:
2569:
2568:
2562:
2558:
2554:
2537:
2524:
2523:
2425:chromatic number
2414:induced subgraph
2402:undirected graph
2354:Mirsky's theorem
2348:Mirsky's theorem
2007:) is an edge in
1931:
1929:
1928:
1923:
1921:
1920:
1902:
1901:
1889:
1888:
1860:
1858:
1857:
1852:
1834:
1832:
1831:
1826:
1814:
1812:
1811:
1806:
1794:
1792:
1791:
1786:
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1728:
1726:
1725:
1720:
1696:
1694:
1693:
1688:
1646:
1644:
1643:
1638:
1636:
1635:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1547:
1545:
1544:
1539:
1521:
1519:
1518:
1513:
1508:
1507:
1482:
1480:
1479:
1474:
1456:
1454:
1453:
1448:
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1384:
1382:
1381:
1376:
1374:
1373:
1357:
1355:
1354:
1349:
1344:
1343:
1325:
1324:
1287:
1285:
1284:
1279:
1267:
1265:
1264:
1259:
1217:
1215:
1214:
1209:
1207:
1206:
1184:
1182:
1181:
1176:
1161:
1159:
1158:
1153:
1141:
1139:
1138:
1133:
1131:
1130:
1118:
1117:
1101:
1099:
1098:
1093:
1081:
1079:
1078:
1073:
1061:
1059:
1058:
1053:
1045:
1044:
1028:
1026:
1025:
1020:
1018:
1017:
1005:
1004:
988:
986:
985:
980:
978:
977:
961:
959:
958:
953:
951:
950:
928:
926:
925:
920:
918:
917:
905:
904:
882:
880:
879:
874:
866:
865:
853:
852:
836:
834:
833:
828:
816:
814:
813:
808:
796:
794:
793:
788:
786:
785:
769:
767:
766:
761:
749:
747:
746:
741:
739:
738:
722:
720:
719:
714:
702:
700:
699:
694:
689:
688:
670:
669:
657:
656:
631:
629:
628:
623:
621:
606:
604:
603:
598:
586:
584:
583:
578:
576:
575:
559:
557:
556:
551:
549:
548:
532:
530:
529:
524:
488:
486:
485:
480:
444:
442:
441:
436:
428:
427:
415:
414:
398:
396:
395:
390:
378:
376:
375:
370:
368:
367:
351:
349:
348:
343:
341:
340:
322:
321:
306:disjoint chains
305:
303:
302:
297:
285:
283:
282:
277:
257:
242:
240:
239:
234:
219:
217:
216:
211:
199:
197:
196:
191:
179:
177:
176:
171:
159:
157:
156:
151:
139:
137:
136:
131:
108:
106:
105:
100:
4045:
4044:
4040:
4039:
4038:
4036:
4035:
4034:
4010:
4009:
4008:
4003:
3999:Young's lattice
3855:
3783:
3722:
3572:Heyting algebra
3520:Boolean algebra
3492:
3473:Laver's theorem
3421:
3387:Boolean algebra
3382:Binary relation
3370:
3347:
3342:
3295:
3294:
3284:
3277:
3270:
3260:
3242:
3234:
3222:
3210:Diaconis, Persi
3200:
3156:
3142:
3114:
3100:10.2307/2316481
3080:
3052:
3038:
3023:
2983:
2982:
2977:
2955:10.2307/2975628
2935:
2921:10.2307/2033375
2901:
2858:
2823:
2809:10.2307/1969503
2789:
2784:
2770:Chvátal, Václav
2764:
2761:
2756:
2755:
2747:
2743:
2735:
2731:
2723:
2719:
2711:
2707:
2699:
2692:
2684:
2680:
2672:
2668:
2660:
2656:
2651:
2635:order dimension
2542:
2532:
2515:
2501:
2500:
2495:
2462:
2453:Boolean lattice
2449:
2410:independent set
2394:
2350:
2344:
2330:
2323:cardinal number
2231:
2207:polynomial time
1954:bipartite graph
1950:KĹ‘nig's theorem
1938:
1912:
1893:
1880:
1863:
1862:
1837:
1836:
1817:
1816:
1797:
1796:
1777:
1776:
1757:
1756:
1731:
1730:
1699:
1698:
1649:
1648:
1627:
1616:
1615:
1596:
1595:
1576:
1575:
1550:
1549:
1524:
1523:
1499:
1485:
1484:
1459:
1458:
1433:
1432:
1413:
1412:
1387:
1386:
1365:
1360:
1359:
1335:
1316:
1290:
1289:
1270:
1269:
1220:
1219:
1198:
1187:
1186:
1167:
1166:
1144:
1143:
1122:
1109:
1104:
1103:
1084:
1083:
1064:
1063:
1036:
1031:
1030:
1009:
996:
991:
990:
969:
964:
963:
942:
931:
930:
909:
896:
885:
884:
857:
844:
839:
838:
819:
818:
799:
798:
777:
772:
771:
752:
751:
730:
725:
724:
705:
704:
680:
661:
648:
634:
633:
614:
609:
608:
589:
588:
567:
562:
561:
540:
535:
534:
491:
490:
447:
446:
419:
406:
401:
400:
381:
380:
359:
354:
353:
332:
313:
308:
307:
288:
287:
250:
245:
244:
225:
224:
202:
201:
182:
181:
162:
161:
142:
141:
122:
121:
91:
90:
87:
85:Inductive proof
69:
21:
12:
11:
5:
4043:
4041:
4033:
4032:
4027:
4025:Perfect graphs
4022:
4012:
4011:
4005:
4004:
4002:
4001:
3996:
3991:
3990:
3989:
3979:
3978:
3977:
3972:
3967:
3957:
3956:
3955:
3945:
3940:
3939:
3938:
3933:
3926:Order morphism
3923:
3922:
3921:
3911:
3906:
3901:
3896:
3891:
3890:
3889:
3879:
3874:
3869:
3863:
3861:
3857:
3856:
3854:
3853:
3852:
3851:
3846:
3844:Locally convex
3841:
3836:
3826:
3824:Order topology
3821:
3820:
3819:
3817:Order topology
3814:
3804:
3794:
3792:
3785:
3784:
3782:
3781:
3776:
3771:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3730:
3728:
3724:
3723:
3721:
3720:
3710:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3664:
3663:
3653:
3648:
3647:
3646:
3641:
3636:
3631:
3629:Chain-complete
3621:
3616:
3615:
3614:
3609:
3604:
3599:
3594:
3584:
3579:
3574:
3569:
3564:
3554:
3549:
3544:
3539:
3534:
3529:
3528:
3527:
3517:
3512:
3506:
3504:
3494:
3493:
3491:
3490:
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3429:
3427:
3423:
3422:
3420:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3378:
3376:
3372:
3371:
3369:
3368:
3363:
3358:
3352:
3349:
3348:
3343:
3341:
3340:
3333:
3326:
3318:
3312:
3311:
3292:
3282:
3258:
3240:
3233:
3232:External links
3230:
3229:
3228:
3198:
3170:(2): 108–109,
3154:
3140:
3112:
3094:(8): 876–877,
3078:
3066:(3): 253–267,
3054:Lovász, László
3050:
3036:
3021:
2990:
2979:Greene, Curtis
2975:
2949:(4): 352–353,
2933:
2915:(4): 701–702,
2899:
2856:
2821:
2803:(1): 161–166,
2787:
2782:
2760:
2757:
2754:
2753:
2741:
2729:
2717:
2705:
2690:
2678:
2674:Fulkerson 1956
2666:
2653:
2652:
2650:
2647:
2585:
2584:
2573:
2567:
2561:
2557:
2553:
2549:
2545:
2541:
2536:
2530:
2527:
2522:
2518:
2514:
2511:
2508:
2491:
2458:
2448:
2445:
2393:
2390:
2346:Main article:
2343:
2340:
2328:
2230:
2227:
2043:and such that
1937:
1934:
1919:
1915:
1911:
1908:
1905:
1900:
1896:
1892:
1887:
1883:
1879:
1876:
1873:
1870:
1850:
1847:
1844:
1824:
1804:
1795:is maximal in
1784:
1764:
1744:
1741:
1738:
1718:
1715:
1712:
1709:
1706:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1634:
1630:
1626:
1623:
1603:
1583:
1563:
1560:
1557:
1537:
1534:
1531:
1511:
1506:
1502:
1498:
1495:
1492:
1472:
1469:
1466:
1446:
1443:
1440:
1420:
1400:
1397:
1394:
1372:
1368:
1347:
1342:
1338:
1334:
1331:
1328:
1323:
1319:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1277:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1205:
1201:
1197:
1194:
1174:
1151:
1129:
1125:
1121:
1116:
1112:
1091:
1071:
1051:
1048:
1043:
1039:
1016:
1012:
1008:
1003:
999:
976:
972:
949:
945:
941:
938:
916:
912:
908:
903:
899:
895:
892:
872:
869:
864:
860:
856:
851:
847:
826:
806:
784:
780:
770:that contains
759:
737:
733:
712:
692:
687:
683:
679:
676:
673:
668:
664:
660:
655:
651:
647:
644:
641:
620:
617:
596:
574:
570:
547:
543:
522:
519:
516:
513:
510:
507:
504:
501:
498:
478:
475:
472:
469:
466:
463:
460:
457:
454:
434:
431:
426:
422:
418:
413:
409:
388:
366:
362:
339:
335:
331:
328:
325:
320:
316:
295:
275:
272:
269:
266:
263:
260:
256:
253:
232:
209:
189:
169:
149:
129:
98:
86:
83:
68:
65:
13:
10:
9:
6:
4:
3:
2:
4042:
4031:
4028:
4026:
4023:
4021:
4018:
4017:
4015:
4000:
3997:
3995:
3992:
3988:
3985:
3984:
3983:
3980:
3976:
3973:
3971:
3968:
3966:
3963:
3962:
3961:
3958:
3954:
3951:
3950:
3949:
3948:Ordered field
3946:
3944:
3941:
3937:
3934:
3932:
3929:
3928:
3927:
3924:
3920:
3917:
3916:
3915:
3912:
3910:
3907:
3905:
3904:Hasse diagram
3902:
3900:
3897:
3895:
3892:
3888:
3885:
3884:
3883:
3882:Comparability
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3864:
3862:
3858:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3831:
3830:
3827:
3825:
3822:
3818:
3815:
3813:
3810:
3809:
3808:
3805:
3803:
3799:
3796:
3795:
3793:
3790:
3786:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3754:Product order
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3731:
3729:
3727:Constructions
3725:
3719:
3715:
3711:
3708:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3662:
3659:
3658:
3657:
3654:
3652:
3649:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3626:
3625:
3624:Partial order
3622:
3620:
3617:
3613:
3612:Join and meet
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3589:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3559:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3526:
3523:
3522:
3521:
3518:
3516:
3513:
3511:
3510:Antisymmetric
3508:
3507:
3505:
3501:
3495:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3428:
3424:
3418:
3417:Weak ordering
3415:
3413:
3410:
3408:
3405:
3403:
3402:Partial order
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3379:
3377:
3373:
3367:
3364:
3362:
3359:
3357:
3354:
3353:
3350:
3346:
3339:
3334:
3332:
3327:
3325:
3320:
3319:
3316:
3307:
3306:
3301:
3298:
3293:
3290:
3289:
3283:
3280:on 2011-07-20
3276:
3269:
3268:
3263:
3262:Babai, László
3259:
3256:on 2007-07-14
3255:
3251:
3250:
3245:
3241:
3239:
3236:
3235:
3231:
3221:
3220:
3215:
3214:Spencer, Joel
3211:
3207:
3206:Aldous, David
3203:
3199:
3195:
3191:
3187:
3183:
3178:
3173:
3169:
3165:
3164:
3159:
3155:
3151:
3147:
3143:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3101:
3097:
3093:
3089:
3088:
3083:
3079:
3074:
3069:
3065:
3061:
3060:
3055:
3051:
3047:
3043:
3039:
3037:0-387-24219-8
3033:
3029:
3028:
3022:
3017:
3012:
3008:
3004:
2988:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2944:
2943:
2938:
2934:
2930:
2926:
2922:
2918:
2914:
2910:
2909:
2904:
2900:
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2867:
2862:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2832:
2827:
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2797:
2792:
2788:
2785:
2779:
2775:
2771:
2767:
2766:Berge, Claude
2763:
2762:
2758:
2750:
2745:
2742:
2738:
2733:
2730:
2726:
2721:
2718:
2714:
2709:
2706:
2702:
2701:Harzheim 2005
2697:
2695:
2691:
2687:
2682:
2679:
2675:
2670:
2667:
2663:
2662:Dilworth 1950
2658:
2655:
2648:
2646:
2643:
2638:
2636:
2632:
2627:
2625:
2621:
2617:
2613:
2609:
2605:
2600:
2598:
2594:
2590:
2571:
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2547:
2539:
2528:
2520:
2516:
2509:
2506:
2499:
2498:
2497:
2494:
2490:
2486:
2482:
2479:}—ordered by
2478:
2474:
2471:-element set
2470:
2466:
2461:
2457:
2454:
2446:
2444:
2442:
2438:
2434:
2433:Lovász (1972)
2430:
2426:
2422:
2417:
2415:
2411:
2407:
2403:
2399:
2391:
2389:
2387:
2383:
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2375:
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2367:
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2359:
2355:
2349:
2339:
2337:
2336:Perles (1963)
2333:
2331:
2324:
2318:
2316:
2312:
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2304:
2300:
2296:
2292:
2288:
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2260:
2256:
2252:
2248:
2244:
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2236:
2228:
2226:
2224:
2220:
2216:
2212:
2208:
2203:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2154:exactly when
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2124:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2090:
2086:
2082:
2078:
2074:
2071:has at least
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1961:
1959:
1955:
1951:
1942:
1933:
1917:
1913:
1909:
1906:
1903:
1898:
1894:
1890:
1885:
1881:
1877:
1871:
1848:
1845:
1842:
1822:
1802:
1782:
1762:
1742:
1739:
1736:
1713:
1707:
1704:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1657:
1654:
1632:
1628:
1624:
1621:
1601:
1581:
1561:
1555:
1535:
1532:
1529:
1504:
1500:
1490:
1470:
1467:
1464:
1444:
1438:
1418:
1398:
1392:
1370:
1366:
1340:
1336:
1332:
1329:
1326:
1321:
1317:
1313:
1310:
1304:
1298:
1288:be the chain
1275:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1228:
1225:
1203:
1199:
1195:
1192:
1172:
1163:
1149:
1127:
1123:
1119:
1114:
1110:
1089:
1069:
1049:
1046:
1041:
1037:
1014:
1010:
1006:
1001:
997:
974:
970:
947:
943:
939:
936:
914:
910:
906:
901:
897:
893:
890:
867:
862:
858:
854:
849:
845:
824:
804:
782:
778:
757:
735:
731:
710:
685:
681:
677:
674:
671:
666:
662:
658:
653:
649:
642:
639:
618:
615:
594:
572:
568:
545:
541:
520:
517:
514:
511:
508:
505:
502:
499:
496:
476:
473:
470:
467:
464:
461:
458:
455:
452:
429:
424:
420:
416:
411:
407:
386:
364:
360:
337:
333:
329:
326:
323:
318:
314:
293:
270:
261:
258:
254:
251:
230:
221:
207:
187:
167:
147:
127:
118:
116:
112:
96:
84:
82:
79:
74:
66:
64:
60:
58:
54:
50:
46:
42:
38:
34:
33:combinatorics
30:
26:
19:
4020:Order theory
3791:& Orders
3769:Star product
3698:Well-founded
3651:Prefix order
3607:Distributive
3597:Complemented
3567:Foundational
3532:Completeness
3488:Zorn's lemma
3447:
3392:Cyclic order
3375:Key concepts
3345:Order theory
3303:
3287:
3275:the original
3266:
3254:the original
3247:
3218:
3167:
3161:
3123:
3091:
3085:
3082:Mirsky, Leon
3063:
3057:
3027:Ordered sets
3026:
3009:(1): 41–68,
3006:
3002:
3001:-families",
2946:
2940:
2937:Galvin, Fred
2912:
2906:
2870:
2864:
2838:(1): 23–32,
2835:
2829:
2800:
2794:
2773:
2744:
2732:
2720:
2708:
2681:
2669:
2657:
2639:
2628:
2623:
2619:
2615:
2611:
2607:
2604:divisibility
2601:
2592:
2588:
2586:
2492:
2488:
2476:
2472:
2468:
2459:
2455:
2450:
2418:
2395:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2351:
2334:
2319:
2314:
2310:
2302:
2298:
2294:
2290:
2286:
2282:
2278:
2274:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2232:
2222:
2218:
2214:
2210:
2204:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2125:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1962:
1947:
1164:
222:
119:
88:
70:
61:
52:
36:
29:order theory
22:
3975:Riesz space
3936:Isomorphism
3812:Normal cone
3734:Composition
3668:Semilattice
3577:Homogeneous
3562:Equivalence
3412:Total order
2737:Steele 1995
2713:Mirsky 1971
2642:antimatroid
1999:and where (
399:. Clearly,
25:mathematics
4014:Categories
3943:Order type
3877:Cofinality
3718:Well-order
3693:Transitive
3582:Idempotent
3515:Asymmetric
3249:PlanetMath
2759:References
2437:complement
2289:has width
2245:has width
632:, and set
3994:Upper set
3931:Embedding
3867:Antichain
3688:Tolerance
3678:Symmetric
3673:Semiorder
3619:Reflexive
3537:Connected
3305:MathWorld
3194:120943065
2852:119826035
2560:⌋
2544:⌊
2510:
2481:inclusion
2465:power set
2146:in which
1907:…
1708:∪
1676:…
1658:∈
1647:for each
1574:. Thus,
1559:∖
1533:−
1494:∖
1468:−
1442:∖
1396:∖
1333:≤
1314:∈
1305:∪
1247:…
1229:∈
1218:for some
1196:≥
940:≤
907:∩
894:∈
871:∅
868:≠
855:∩
675:…
515:…
471:…
433:∅
430:≠
417:∩
327:…
265:∖
73:antichain
67:Statement
45:antichain
3789:Topology
3656:Preorder
3639:Eulerian
3602:Complete
3552:Directed
3542:Covering
3407:Preorder
3366:Category
3361:Glossary
3264:(2005),
2772:(1984),
2267:coloring
1987:) where
1625:≱
1120:≱
1047:≱
1029:, since
1007:≱
619:′
379:of size
255:′
78:disjoint
3894:Duality
3872:Cofinal
3860:Related
3839:Fréchet
3716:)
3592:Bounded
3587:Lattice
3560:)
3558:Partial
3426:Results
3397:Lattice
3186:0168497
3150:2920058
3108:2316481
3046:2127991
2971:1270960
2963:2975628
2929:2033375
2895:1363140
2887:2079151
2817:1969503
2463:is the
2421:perfect
2269:of the
2111:; then
2067:; then
2055:. Let
1861:chains
1815:). Now
1775:(since
1697:, then
929:. Then
837:. Then
113: (
3919:Subnet
3899:Filter
3849:Normed
3834:Banach
3800:&
3707:Better
3644:Strict
3634:Graded
3525:topics
3356:Topics
3192:
3184:
3148:
3138:
3106:
3044:
3034:
2969:
2961:
2927:
2893:
2885:
2850:
2815:
2780:
2467:of an
2435:, the
2406:clique
2400:is an
2301:has a
2166:is in
2158:is in
1268:. Let
883:. Let
533:, let
489:. For
111:Galvin
49:chains
3909:Ideal
3887:Graph
3683:Total
3661:Total
3547:Dense
3278:(PDF)
3271:(PDF)
3223:(PDF)
3190:S2CID
3104:JSTOR
2959:JSTOR
2925:JSTOR
2891:S2CID
2866:Order
2848:S2CID
2831:Order
2813:JSTOR
2649:Notes
2637:two.
2507:width
2174:from
2150:<
2107:) in
2015:<
2011:when
1967:with
53:width
3500:list
3136:ISBN
3032:ISBN
2778:ISBN
2629:The
2451:The
2225:).
2115:has
2095:and
2047:and
1082:and
817:and
445:for
120:Let
115:1994
31:and
3914:Net
3714:Pre
3172:doi
3128:doi
3096:doi
3068:doi
3011:doi
2951:doi
2947:101
2917:doi
2875:doi
2840:doi
2805:doi
2431:of
2297:of
2273:of
2257:of
2178:to
2130:= (
2035:in
2027:in
2019:in
1975:= (
1952:on
1755:in
1548:in
607:in
117:).
71:An
23:In
4016::
3302:.
3246:,
3212:;
3208:;
3188:,
3182:MR
3180:,
3166:,
3146:MR
3144:,
3134:,
3118:;
3102:,
3092:78
3090:,
3062:,
3042:MR
3040:,
3007:20
3005:,
2967:MR
2965:,
2957:,
2945:,
2923:,
2911:,
2889:,
2883:MR
2881:,
2871:20
2869:,
2863:,
2846:,
2834:,
2811:,
2801:51
2799:,
2768:;
2693:^
2626:.
2396:A
2309:,
2223:kn
2162:,
2119:-
2075:-
1995:=
1991:=
1960:.
1385:,
643::=
259::=
220:.
35:,
3712:(
3709:)
3705:(
3556:(
3503:)
3337:e
3330:t
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3308:.
3227:.
3197:.
3174::
3168:1
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3077:.
3070::
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3013::
2989:k
2974:.
2953::
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2919::
2913:7
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2807::
2751:.
2739:.
2727:.
2715:.
2703:.
2688:.
2676:.
2664:.
2624:n
2620:m
2616:m
2612:n
2608:n
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2589:X
2572:.
2566:)
2556:2
2552:/
2548:n
2540:n
2535:(
2529:=
2526:)
2521:n
2517:B
2513:(
2493:n
2489:B
2477:n
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2469:n
2460:n
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2380:(
2378:N
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2368:(
2366:N
2362:x
2358:x
2329:0
2327:ℵ
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2303:w
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2009:G
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1993:V
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1985:E
1983:,
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1973:G
1969:n
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
