1952:
2331:
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.
73:
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.
91:
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
2655:
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.
86:
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
2593:
2363:
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
1367:
2450:
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 (
712:
1941:
892:
454:
295:
1706:
1277:
1531:
1151:
1038:
938:
2201:
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
361:
1656:
1071:
542:
498:
1583:
1466:
1420:
1738:
1227:
971:
2918:
641:
1394:
998:
806:
759:
596:
569:
388:
2438:
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
1870:
1764:
1557:
1492:
2317:
3010:
2607:
1844:
1824:
1804:
1784:
1623:
1603:
1440:
1297:
1194:
1171:
1111:
1091:
846:
826:
779:
732:
616:
408:
315:
252:
229:
209:
189:
169:
149:
118:
1955:
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
3997:
3276:
2513:
3980:
3510:
3346:
3150:
2792:
3254:
3827:
1960:
2427:
of a comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements.
2399:, is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain.
2415:
formed from a partial order by creating a vertex per element of the order, and an edge connecting any two comparable elements. Thus, a
4040:
3963:
3822:
3046:
3817:
3453:
3535:
3069:
2420:
2641:
3854:
3774:
3173:
3097:
2952:
1302:
3448:
3639:
3568:
2836:
3542:
3530:
3493:
3468:
3443:
3397:
3366:
3473:
3463:
646:
2296:
colors; every color class in a proper coloring of the incomparability graph must be a chain. By the assumption that
1875:
851:
413:
3839:
3339:
1968:
3812:
3478:
2454:). Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.
3744:
3371:
3130:
28:
3285:
2094:
contains vertices corresponding to the same element on both sides of the bipartition) and no two elements of
4035:
3992:
3975:
3228:
3126:
2244:
Dilworth's theorem for infinite partially ordered sets states that a partially ordered set has finite width
2134:
chains. Therefore, we have constructed an antichain and a partition into chains with the same cardinality.
1661:
1232:
3904:
3520:
2491:
1497:
1116:
1003:
897:
257:
3064:
2644:
on monotone subsequences can be interpreted as an application of
Dilworth's theorem to partial orders of
4030:
3882:
3717:
3708:
3577:
3412:
3376:
3332:
2806:
2439:
2416:
320:
51:
1628:
1043:
3970:
3929:
3919:
3909:
3654:
3617:
3607:
3587:
3572:
3041:, Advances in Mathematics (Springer), vol. 7, New York: Springer, Theorem 5.6, p. 60,
2780:
2495:
59:
503:
459:
3897:
3808:
3754:
3713:
3703:
3592:
3525:
3488:
2408:
2364:
2358:
2281:
3264:
3236:, IMA Volumes in Mathematics and its Applications, vol. 72, Springer-Verlag, pp. 111–131
1562:
1445:
1399:
3936:
3789:
3698:
3688:
3629:
3547:
3297:
3200:
3114:
2969:
2935:
2901:
2858:
2823:
2801:
1711:
1199:
943:
67:
4009:
3849:
3483:
3272:
3946:
3924:
3784:
3769:
3749:
3552:
3307:
3212:
3146:
3042:
2788:
2216:
This connection to bipartite matching allows the width of any partial order to be computed in
3036:
3759:
3612:
3182:
3138:
3106:
3078:
3021:
2961:
2927:
2885:
2850:
2815:
2447:
2435:
2424:
2412:
3196:
3160:
3056:
2981:
2897:
1372:
976:
784:
737:
574:
547:
366:
3941:
3724:
3602:
3597:
3582:
3407:
3392:
3192:
3156:
3052:
2977:
2913:
2893:
2876:
2841:
2645:
2463:
2333:
2217:
1964:
3498:
3248:
2889:
2367:. Its proof is much simpler than the proof of Dilworth's theorem itself: for any element
1849:
1743:
1536:
1471:
1951:
621:
3859:
3844:
3834:
3693:
3671:
3649:
3220:
2995:
2277:
1829:
1809:
1789:
1769:
1608:
1588:
1425:
1282:
1179:
1156:
1096:
1076:
831:
811:
764:
717:
601:
393:
300:
237:
214:
194:
174:
154:
134:
103:
2872:"Recognition algorithms for orders of small width and graphs of small Dilworth number"
92:
partial order is defined as the common size of the antichain and chain decomposition.
4024:
3958:
3914:
3892:
3764:
3634:
3622:
3427:
3204:
3083:
3026:
2989:
2862:
2431:
1959:
Like a number of other results in combinatorics, Dilworth's theorem is equivalent to
43:
3310:
3299:
Recognition
Algorithms for Orders of Small Width and Graphs of Small Dilworth Number
2871:
2610:
also concerns antichains in a power set and can be used to prove
Sperner's theorem.
3779:
3661:
3644:
3562:
3402:
3355:
3224:
3216:
3168:
2905:
2776:
2614:
2337:
88:
39:
17:
2839:(1988), "Combinatorial representation and convex dimension of convex geometries",
2328:-colorable incomparability graph, and thus has the desired partition into chains.
3215:(1995), "Variations on the monotone subsequence theme of Erdős and Szekeres", in
3985:
3678:
3557:
3422:
3137:, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 42,
3092:
2947:
2652:
2588:{\displaystyle \operatorname {width} (B_{n})={n \choose \lfloor {n/2}\rfloor }.}
121:
35:
2916:(1956), "Note on Dilworth's decomposition theorem for partially ordered sets",
3953:
3887:
3728:
3259:
3142:
4004:
3877:
3683:
3315:
2475:
83:
55:
3278:
Lecture Notes in
Combinatorics and Probability, Lecture 10: Perfect Graphs
100:
The following proof by induction on the size of the partially ordered set
3799:
3666:
3417:
2292:
as vertices, with an edge between every two incomparable elements) using
2787:, Annals of Discrete Mathematics, vol. 21, Elsevier, p. viii,
2633:
2. Therefore, by
Dilworth's theorem, the width of this partial order is
2137:
To prove KĹ‘nig's theorem from
Dilworth's theorem, for a bipartite graph
3187:
3118:
2973:
2939:
2854:
2827:
2304:, and by the finite version of Dilworth's theorem, every finite subset
27:"Chain decomposition" redirects here. For the path decomposition, see
3110:
2965:
2931:
2819:
2349:
discusses analogues of Dilworth's theorem in the infinite setting.
2050:, such that each edge in the graph contains at least one vertex in
66:
of the partially order. The theorem is named for the mathematician
1950:
151:
be a finite partially ordered set. The theorem holds trivially if
3328:
3324:
3067:(1972), "Normal hypergraphs and the perfect graph conjecture",
2804:(1950), "A Decomposition Theorem for Partially Ordered Sets",
2950:(1994), "A proof of Dilworth's chain decomposition theorem",
2598:
In other words, a largest family of incomparable subsets of
2870:
Felsner, Stefan; Raghavan, Vijay; Spinrad, Jeremy (2003),
2423:
in a comparability graph corresponds to an antichain. Any
1982:
elements, using KĹ‘nig's theorem, define a bipartite graph
2992:; Kleitman, Daniel J. (1976), "The structure of Sperner
2419:
in a comparability graph corresponds to a chain, and an
62:
needed to cover all elements. This number is called the
2617:, the subinterval forms an antichain with cardinality
3171:(1963), "On Dilworth's theorem in the infinite case",
3095:(1971), "A dual of Dilworth's decomposition theorem",
2707:
2705:
2696:
2336:Îş there is an infinite partially ordered set of width
1967:
matching and several other related theorems including
58:
of incomparable elements equals the minimum number of
2998:
2516:
2343:
whose partition into the fewest chains has Îş chains.
2252:
chains. For, suppose that an infinite partial order
1878:
1852:
1832:
1812:
1792:
1772:
1746:
1714:
1664:
1631:
1611:
1591:
1565:
1539:
1500:
1474:
1448:
1428:
1402:
1375:
1305:
1285:
1235:
1202:
1182:
1159:
1119:
1099:
1079:
1046:
1006:
979:
946:
900:
854:
834:
814:
787:
767:
740:
720:
649:
624:
604:
577:
550:
506:
462:
416:
396:
369:
323:
303:
260:
240:
217:
197:
177:
157:
137:
106:
3296:
Felsner, S.; Raghavan, V. & Spinrad, J. (1999),
3249:
Equivalence of seven major theorems in combinatorics
2395:), consisting of elements that have equal values of
2316:-colorable incomparability graph. Therefore, by the
1362:{\displaystyle \{a\}\cup \{z\in C_{i}:z\leq x_{i}\}}
3870:
3798:
3737:
3507:
3436:
3385:
2193:. By Dilworth's theorem, there exists an antichain
3004:
2587:
1935:
1864:
1838:
1818:
1798:
1778:
1758:
1732:
1700:
1650:
1617:
1597:
1577:
1551:
1525:
1486:
1460:
1434:
1414:
1388:
1361:
1291:
1271:
1221:
1188:
1165:
1145:
1105:
1085:
1065:
1032:
992:
965:
932:
886:
840:
820:
800:
773:
753:
726:
706:
635:
610:
590:
563:
536:
492:
448:
402:
382:
355:
309:
289:
246:
223:
203:
183:
163:
143:
112:
2576:
2545:
2260:, meaning that there are at most a finite number
2919:Proceedings of the American Mathematical Society
2625:chains is easy to achieve: for each odd integer
2205:form a matching in the graph. The complement of
1974:To prove Dilworth's theorem for a partial order
2034:. By KĹ‘nig's theorem, there exists a matching
707:{\displaystyle A:=\{x_{1},x_{2},\dots ,x_{k}\}}
1936:{\displaystyle \{a\},C_{1},C_{2},\dots ,C_{k}}
887:{\displaystyle A_{i}\cap C_{j}\neq \emptyset }
449:{\displaystyle A_{0}\cap C_{i}\neq \emptyset }
234:By induction, we assume that for some integer
3340:
2629:in , form a chain of the numbers of the form
2613:If we order the integers in the interval by
2353:Dual of Dilworth's theorem (Mirsky's theorem)
2264:of elements in any antichain. For any subset
2110:in the same chain whenever there is an edge (
8:
3135:Sparsity: Graphs, Structures, and Algorithms
2735:
2570:
2554:
2451:
2276:chains (if it exists) may be described as a
2240:Extension to infinite partially ordered sets
2213:with the same cardinality as this matching.
1885:
1879:
1727:
1721:
1695:
1671:
1520:
1507:
1356:
1318:
1312:
1306:
1266:
1242:
701:
656:
284:
278:
2759:
2153:), form a partial order on the vertices of
3998:Positive cone of a partially ordered group
3347:
3333:
3325:
2383:) denote the size of the largest of these
2352:
2248:if and only if it may be partitioned into
2102:be a family of chains formed by including
3186:
3082:
3025:
3014:Journal of Combinatorial Theory, Series A
2997:
2684:
2621:. A partition of this partial order into
2575:
2561:
2557:
2544:
2542:
2530:
2515:
1927:
1908:
1895:
1877:
1851:
1831:
1811:
1791:
1771:
1745:
1713:
1663:
1642:
1630:
1610:
1590:
1564:
1538:
1514:
1499:
1473:
1447:
1427:
1401:
1380:
1374:
1350:
1331:
1304:
1284:
1234:
1213:
1201:
1181:
1158:
1137:
1124:
1118:
1098:
1078:
1051:
1045:
1024:
1011:
1005:
984:
978:
957:
945:
924:
911:
899:
872:
859:
853:
833:
813:
792:
786:
766:
745:
739:
719:
695:
676:
663:
648:
623:
603:
582:
576:
555:
549:
505:
461:
434:
421:
415:
395:
374:
368:
347:
328:
322:
302:
259:
239:
216:
196:
176:
156:
136:
105:
3981:Positive cone of an ordered vector space
2711:
2672:
2602:is obtained by selecting the subsets of
2074:that do not correspond to any vertex in
2665:
1625:disjoint chains, as required. Next, if
1569:
1504:
1452:
1406:
275:
2747:
2723:
2443:
2346:
125:
1701:{\displaystyle i\in \{1,2,\dots ,k\}}
1272:{\displaystyle i\in \{1,2,\dots ,k\}}
598:that belongs to an antichain of size
7:
2697:Felsner, Raghavan & Spinrad 2003
2608:Lubell–Yamamoto–Meshalkin inequality
1526:{\displaystyle A\setminus \{x_{i}\}}
1146:{\displaystyle x_{j}\not \geq x_{i}}
1033:{\displaystyle x_{i}\not \geq x_{j}}
933:{\displaystyle y\in A_{i}\cap C_{j}}
290:{\displaystyle P':=P\setminus \{a\}}
3230:Discrete Probability and Algorithms
2498:states that a maximum antichain of
2434:if, in every induced subgraph, the
1422:does not have an antichain of size
3508:Properties & Types (
2890:10.1023/B:ORDE.0000034609.99940.fb
2549:
2403:Perfection of comparability graphs
2375:as their largest element, and let
2332:In particular, for every infinite
2288:(a graph that has the elements of
2098:are comparable to each other. Let
881:
443:
356:{\displaystyle C_{1},\dots ,C_{k}}
191:has at least one element, and let
25:
3964:Positive cone of an ordered field
2953:The American Mathematical Monthly
2224:-element partial orders of width
808:. Fix arbitrary distinct indices
3818:Ordered topological vector space
2371:, consider the chains that have
1651:{\displaystyle a\not \geq x_{i}}
1073:. By interchanging the roles of
1066:{\displaystyle x_{i}\not \geq y}
2458:Width of special partial orders
2387:-maximal chains. Then each set
363:and has at least one antichain
2536:
2523:
2181:, and there exists an edge in
1442:. Induction then implies that
1113:in this argument we also have
537:{\displaystyle i=1,2,\dots ,k}
493:{\displaystyle i=1,2,\dots ,k}
1:
3775:Series-parallel partial order
3174:Israel Journal of Mathematics
3098:American Mathematical Monthly
2651:The "convex dimension" of an
3454:Cantor's isomorphism theorem
3255:"Dual of Dilworth's Theorem"
3084:10.1016/0012-365X(72)90006-4
3027:10.1016/0097-3165(76)90077-7
2197:and a partition into chains
1578:{\displaystyle P\setminus K}
1461:{\displaystyle P\setminus K}
1415:{\displaystyle P\setminus K}
70:, who published it in 1950.
3494:Szpilrajn extension theorem
3469:Hausdorff maximal principle
3444:Boolean prime ideal theorem
2606:that have median size. The
2090:elements (possibly more if
1733:{\displaystyle A\cup \{a\}}
1222:{\displaystyle a\geq x_{i}}
966:{\displaystyle y\leq x_{j}}
50:states that, in any finite
4057:
3840:Topological vector lattice
2494:or, notationally, (2, ⊆).
2356:
2228:can be recognized in time
2070:be the set of elements of
2062:have the same cardinality
571:be the maximal element in
254:the partially ordered set
171:is empty. So, assume that
26:
4041:Theorems in combinatorics
3362:
3143:10.1007/978-3-642-27875-4
3131:Ossona de Mendez, Patrice
3035:Harzheim, Egbert (2005),
1947:Proof via KĹ‘nig's theorem
54:, the maximum size of an
3449:Cantor–Bernstein theorem
3133:(2012), "Theorem 3.13",
2785:Topics on Perfect Graphs
2736:Berge & Chvátal 1984
2452:Berge & Chvátal 1984
2209:forms a vertex cover in
2042:, and a set of vertices
1943:, completing the proof.
1740:is an antichain of size
1533:is an antichain of size
1369:. Then by the choice of
761:be an antichain of size
211:be a maximal element of
29:Heavy path decomposition
3993:Partially ordered group
3813:Specialization preorder
2760:Edelman & Saks 1988
2486:—essentially {1, 2, …,
2430:An undirected graph is
2318:De Bruijn–Erdős theorem
2272:, a decomposition into
1969:Hall's marriage theorem
1946:
973:, by the definition of
3479:Kruskal's tree theorem
3474:Knaster–Tarski theorem
3464:Dushnik–Miller theorem
3227:; et al. (eds.),
3006:
2642:Erdős–Szekeres theorem
2589:
1956:
1937:
1866:
1846:can be covered by the
1840:
1820:
1800:
1780:
1760:
1734:
1702:
1652:
1619:
1599:
1579:
1553:
1527:
1494:disjoint chains since
1488:
1462:
1436:
1416:
1390:
1363:
1293:
1273:
1223:
1190:
1167:
1147:
1107:
1087:
1067:
1034:
994:
967:
934:
888:
842:
822:
802:
775:
755:
734:is an antichain. Let
728:
708:
637:
612:
592:
565:
538:
494:
450:
404:
384:
357:
311:
291:
248:
225:
205:
185:
165:
145:
114:
3007:
2884:(4): 351–364 (2004),
2807:Annals of Mathematics
2590:
2440:perfect graph theorem
2282:incomparability graph
1954:
1938:
1867:
1841:
1821:
1801:
1781:
1761:
1735:
1703:
1653:
1620:
1600:
1580:
1554:
1528:
1489:
1463:
1437:
1417:
1391:
1389:{\displaystyle x_{i}}
1364:
1294:
1274:
1224:
1196:. Suppose first that
1191:
1168:
1153:. This verifies that
1148:
1108:
1088:
1068:
1035:
995:
993:{\displaystyle x_{j}}
968:
935:
889:
843:
823:
803:
801:{\displaystyle x_{i}}
776:
756:
754:{\displaystyle A_{i}}
729:
709:
638:
613:
593:
591:{\displaystyle C_{i}}
566:
564:{\displaystyle x_{i}}
539:
495:
451:
405:
385:
383:{\displaystyle A_{0}}
358:
312:
292:
249:
226:
206:
186:
166:
146:
115:
52:partially ordered set
3971:Ordered vector space
3070:Discrete Mathematics
2996:
2514:
1876:
1850:
1830:
1810:
1790:
1770:
1744:
1712:
1662:
1629:
1609:
1589:
1563:
1537:
1498:
1472:
1446:
1426:
1400:
1373:
1303:
1283:
1233:
1200:
1180:
1157:
1117:
1097:
1077:
1044:
1004:
1000:. This implies that
977:
944:
898:
852:
832:
812:
785:
765:
738:
718:
647:
622:
602:
575:
548:
504:
460:
414:
394:
367:
321:
301:
258:
238:
215:
195:
175:
155:
135:
120:is based on that of
104:
3809:Alexandrov topology
3755:Lexicographic order
3714:Well-quasi-ordering
2802:Dilworth, Robert P.
2409:comparability graph
1865:{\displaystyle k+1}
1759:{\displaystyle k+1}
1552:{\displaystyle k-1}
1487:{\displaystyle k-1}
18:Chain decomposition
3790:Transitive closure
3750:Converse/Transpose
3459:Dilworth's theorem
3311:"Dilworth's Lemma"
3308:Weisstein, Eric W.
3213:Steele, J. Michael
3188:10.1007/BF02759806
3127:Nešetřil, Jaroslav
3002:
2855:10.1007/BF00143895
2835:Edelman, Paul H.;
2585:
2324:itself also has a
2220:. More precisely,
1957:
1933:
1862:
1836:
1816:
1796:
1776:
1756:
1730:
1698:
1648:
1615:
1605:can be covered by
1595:
1575:
1549:
1523:
1484:
1468:can be covered by
1458:
1432:
1412:
1386:
1359:
1289:
1269:
1219:
1186:
1163:
1143:
1103:
1083:
1063:
1030:
990:
963:
930:
884:
838:
818:
798:
771:
751:
724:
704:
636:{\displaystyle P'}
633:
608:
588:
561:
534:
490:
446:
400:
380:
353:
307:
297:can be covered by
287:
244:
221:
201:
181:
161:
141:
110:
68:Robert P. Dilworth
48:Dilworth's theorem
38:, in the areas of
4018:
4017:
3976:Partially ordered
3785:Symmetric closure
3770:Reflexive closure
3513:
3152:978-3-642-27874-7
3005:{\displaystyle k}
2794:978-0-444-86587-8
2574:
2507:has size at most
2496:Sperner's theorem
1839:{\displaystyle P}
1819:{\displaystyle P}
1799:{\displaystyle a}
1779:{\displaystyle P}
1618:{\displaystyle k}
1598:{\displaystyle P}
1435:{\displaystyle k}
1292:{\displaystyle K}
1189:{\displaystyle P}
1176:We now return to
1173:is an antichain.
1166:{\displaystyle A}
1106:{\displaystyle j}
1086:{\displaystyle i}
841:{\displaystyle j}
821:{\displaystyle i}
774:{\displaystyle k}
727:{\displaystyle A}
714:. We claim that
611:{\displaystyle k}
403:{\displaystyle k}
310:{\displaystyle k}
247:{\displaystyle k}
224:{\displaystyle P}
204:{\displaystyle a}
184:{\displaystyle P}
164:{\displaystyle P}
144:{\displaystyle P}
113:{\displaystyle P}
16:(Redirected from
4048:
3760:Linear extension
3509:
3489:Mirsky's theorem
3349:
3342:
3335:
3326:
3321:
3320:
3302:
3292:
3290:
3284:, archived from
3283:
3268:
3263:, archived from
3237:
3235:
3207:
3190:
3169:Perles, Micha A.
3163:
3121:
3087:
3086:
3059:
3030:
3029:
3011:
3009:
3008:
3003:
2984:
2942:
2914:Fulkerson, D. R.
2908:
2865:
2837:Saks, Michael E.
2830:
2797:
2763:
2757:
2751:
2745:
2739:
2733:
2727:
2721:
2715:
2709:
2700:
2694:
2688:
2682:
2676:
2670:
2594:
2592:
2591:
2586:
2581:
2580:
2579:
2573:
2569:
2565:
2548:
2535:
2534:
2436:chromatic number
2425:induced subgraph
2413:undirected graph
2365:Mirsky's theorem
2359:Mirsky's theorem
2018:) is an edge in
1942:
1940:
1939:
1934:
1932:
1931:
1913:
1912:
1900:
1899:
1871:
1869:
1868:
1863:
1845:
1843:
1842:
1837:
1825:
1823:
1822:
1817:
1805:
1803:
1802:
1797:
1785:
1783:
1782:
1777:
1765:
1763:
1762:
1757:
1739:
1737:
1736:
1731:
1707:
1705:
1704:
1699:
1657:
1655:
1654:
1649:
1647:
1646:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1558:
1556:
1555:
1550:
1532:
1530:
1529:
1524:
1519:
1518:
1493:
1491:
1490:
1485:
1467:
1465:
1464:
1459:
1441:
1439:
1438:
1433:
1421:
1419:
1418:
1413:
1395:
1393:
1392:
1387:
1385:
1384:
1368:
1366:
1365:
1360:
1355:
1354:
1336:
1335:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1228:
1226:
1225:
1220:
1218:
1217:
1195:
1193:
1192:
1187:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1142:
1141:
1129:
1128:
1112:
1110:
1109:
1104:
1092:
1090:
1089:
1084:
1072:
1070:
1069:
1064:
1056:
1055:
1039:
1037:
1036:
1031:
1029:
1028:
1016:
1015:
999:
997:
996:
991:
989:
988:
972:
970:
969:
964:
962:
961:
939:
937:
936:
931:
929:
928:
916:
915:
893:
891:
890:
885:
877:
876:
864:
863:
847:
845:
844:
839:
827:
825:
824:
819:
807:
805:
804:
799:
797:
796:
780:
778:
777:
772:
760:
758:
757:
752:
750:
749:
733:
731:
730:
725:
713:
711:
710:
705:
700:
699:
681:
680:
668:
667:
642:
640:
639:
634:
632:
617:
615:
614:
609:
597:
595:
594:
589:
587:
586:
570:
568:
567:
562:
560:
559:
543:
541:
540:
535:
499:
497:
496:
491:
455:
453:
452:
447:
439:
438:
426:
425:
409:
407:
406:
401:
389:
387:
386:
381:
379:
378:
362:
360:
359:
354:
352:
351:
333:
332:
317:disjoint chains
316:
314:
313:
308:
296:
294:
293:
288:
268:
253:
251:
250:
245:
230:
228:
227:
222:
210:
208:
207:
202:
190:
188:
187:
182:
170:
168:
167:
162:
150:
148:
147:
142:
119:
117:
116:
111:
21:
4056:
4055:
4051:
4050:
4049:
4047:
4046:
4045:
4021:
4020:
4019:
4014:
4010:Young's lattice
3866:
3794:
3733:
3583:Heyting algebra
3531:Boolean algebra
3503:
3484:Laver's theorem
3432:
3398:Boolean algebra
3393:Binary relation
3381:
3358:
3353:
3306:
3305:
3295:
3288:
3281:
3271:
3253:
3245:
3233:
3221:Diaconis, Persi
3211:
3167:
3153:
3125:
3111:10.2307/2316481
3091:
3063:
3049:
3034:
2994:
2993:
2988:
2966:10.2307/2975628
2946:
2932:10.2307/2033375
2912:
2869:
2834:
2820:10.2307/1969503
2800:
2795:
2781:Chvátal, Václav
2775:
2772:
2767:
2766:
2758:
2754:
2746:
2742:
2734:
2730:
2722:
2718:
2710:
2703:
2695:
2691:
2683:
2679:
2671:
2667:
2662:
2646:order dimension
2553:
2543:
2526:
2512:
2511:
2506:
2473:
2464:Boolean lattice
2460:
2421:independent set
2405:
2361:
2355:
2341:
2334:cardinal number
2242:
2218:polynomial time
1965:bipartite graph
1961:KĹ‘nig's theorem
1949:
1923:
1904:
1891:
1874:
1873:
1848:
1847:
1828:
1827:
1808:
1807:
1788:
1787:
1768:
1767:
1742:
1741:
1710:
1709:
1660:
1659:
1638:
1627:
1626:
1607:
1606:
1587:
1586:
1561:
1560:
1535:
1534:
1510:
1496:
1495:
1470:
1469:
1444:
1443:
1424:
1423:
1398:
1397:
1376:
1371:
1370:
1346:
1327:
1301:
1300:
1281:
1280:
1231:
1230:
1209:
1198:
1197:
1178:
1177:
1155:
1154:
1133:
1120:
1115:
1114:
1095:
1094:
1075:
1074:
1047:
1042:
1041:
1020:
1007:
1002:
1001:
980:
975:
974:
953:
942:
941:
920:
907:
896:
895:
868:
855:
850:
849:
830:
829:
810:
809:
788:
783:
782:
763:
762:
741:
736:
735:
716:
715:
691:
672:
659:
645:
644:
625:
620:
619:
600:
599:
578:
573:
572:
551:
546:
545:
502:
501:
458:
457:
430:
417:
412:
411:
392:
391:
370:
365:
364:
343:
324:
319:
318:
299:
298:
261:
256:
255:
236:
235:
213:
212:
193:
192:
173:
172:
153:
152:
133:
132:
102:
101:
98:
96:Inductive proof
80:
32:
23:
22:
15:
12:
11:
5:
4054:
4052:
4044:
4043:
4038:
4036:Perfect graphs
4033:
4023:
4022:
4016:
4015:
4013:
4012:
4007:
4002:
4001:
4000:
3990:
3989:
3988:
3983:
3978:
3968:
3967:
3966:
3956:
3951:
3950:
3949:
3944:
3937:Order morphism
3934:
3933:
3932:
3922:
3917:
3912:
3907:
3902:
3901:
3900:
3890:
3885:
3880:
3874:
3872:
3868:
3867:
3865:
3864:
3863:
3862:
3857:
3855:Locally convex
3852:
3847:
3837:
3835:Order topology
3832:
3831:
3830:
3828:Order topology
3825:
3815:
3805:
3803:
3796:
3795:
3793:
3792:
3787:
3782:
3777:
3772:
3767:
3762:
3757:
3752:
3747:
3741:
3739:
3735:
3734:
3732:
3731:
3721:
3711:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3675:
3674:
3664:
3659:
3658:
3657:
3652:
3647:
3642:
3640:Chain-complete
3632:
3627:
3626:
3625:
3620:
3615:
3610:
3605:
3595:
3590:
3585:
3580:
3575:
3565:
3560:
3555:
3550:
3545:
3540:
3539:
3538:
3528:
3523:
3517:
3515:
3505:
3504:
3502:
3501:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3451:
3446:
3440:
3438:
3434:
3433:
3431:
3430:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3389:
3387:
3383:
3382:
3380:
3379:
3374:
3369:
3363:
3360:
3359:
3354:
3352:
3351:
3344:
3337:
3329:
3323:
3322:
3303:
3293:
3269:
3251:
3244:
3243:External links
3241:
3240:
3239:
3209:
3181:(2): 108–109,
3165:
3151:
3123:
3105:(8): 876–877,
3089:
3077:(3): 253–267,
3065:Lovász, László
3061:
3047:
3032:
3001:
2990:Greene, Curtis
2986:
2960:(4): 352–353,
2944:
2926:(4): 701–702,
2910:
2867:
2832:
2814:(1): 161–166,
2798:
2793:
2771:
2768:
2765:
2764:
2752:
2740:
2728:
2716:
2701:
2689:
2685:Fulkerson 1956
2677:
2664:
2663:
2661:
2658:
2596:
2595:
2584:
2578:
2572:
2568:
2564:
2560:
2556:
2552:
2547:
2541:
2538:
2533:
2529:
2525:
2522:
2519:
2502:
2469:
2459:
2456:
2404:
2401:
2357:Main article:
2354:
2351:
2339:
2241:
2238:
2054:and such that
1948:
1945:
1930:
1926:
1922:
1919:
1916:
1911:
1907:
1903:
1898:
1894:
1890:
1887:
1884:
1881:
1861:
1858:
1855:
1835:
1815:
1806:is maximal in
1795:
1775:
1755:
1752:
1749:
1729:
1726:
1723:
1720:
1717:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1645:
1641:
1637:
1634:
1614:
1594:
1574:
1571:
1568:
1548:
1545:
1542:
1522:
1517:
1513:
1509:
1506:
1503:
1483:
1480:
1477:
1457:
1454:
1451:
1431:
1411:
1408:
1405:
1383:
1379:
1358:
1353:
1349:
1345:
1342:
1339:
1334:
1330:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1288:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1216:
1212:
1208:
1205:
1185:
1162:
1140:
1136:
1132:
1127:
1123:
1102:
1082:
1062:
1059:
1054:
1050:
1027:
1023:
1019:
1014:
1010:
987:
983:
960:
956:
952:
949:
927:
923:
919:
914:
910:
906:
903:
883:
880:
875:
871:
867:
862:
858:
837:
817:
795:
791:
781:that contains
770:
748:
744:
723:
703:
698:
694:
690:
687:
684:
679:
675:
671:
666:
662:
658:
655:
652:
631:
628:
607:
585:
581:
558:
554:
533:
530:
527:
524:
521:
518:
515:
512:
509:
489:
486:
483:
480:
477:
474:
471:
468:
465:
445:
442:
437:
433:
429:
424:
420:
399:
377:
373:
350:
346:
342:
339:
336:
331:
327:
306:
286:
283:
280:
277:
274:
271:
267:
264:
243:
220:
200:
180:
160:
140:
109:
97:
94:
79:
76:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4053:
4042:
4039:
4037:
4034:
4032:
4029:
4028:
4026:
4011:
4008:
4006:
4003:
3999:
3996:
3995:
3994:
3991:
3987:
3984:
3982:
3979:
3977:
3974:
3973:
3972:
3969:
3965:
3962:
3961:
3960:
3959:Ordered field
3957:
3955:
3952:
3948:
3945:
3943:
3940:
3939:
3938:
3935:
3931:
3928:
3927:
3926:
3923:
3921:
3918:
3916:
3915:Hasse diagram
3913:
3911:
3908:
3906:
3903:
3899:
3896:
3895:
3894:
3893:Comparability
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3875:
3873:
3869:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3842:
3841:
3838:
3836:
3833:
3829:
3826:
3824:
3821:
3820:
3819:
3816:
3814:
3810:
3807:
3806:
3804:
3801:
3797:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3766:
3765:Product order
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3742:
3740:
3738:Constructions
3736:
3730:
3726:
3722:
3719:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3673:
3670:
3669:
3668:
3665:
3663:
3660:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3637:
3636:
3635:Partial order
3633:
3631:
3628:
3624:
3623:Join and meet
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3600:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3570:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3537:
3534:
3533:
3532:
3529:
3527:
3524:
3522:
3521:Antisymmetric
3519:
3518:
3516:
3512:
3506:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3441:
3439:
3435:
3429:
3428:Weak ordering
3426:
3424:
3421:
3419:
3416:
3414:
3413:Partial order
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3390:
3388:
3384:
3378:
3375:
3373:
3370:
3368:
3365:
3364:
3361:
3357:
3350:
3345:
3343:
3338:
3336:
3331:
3330:
3327:
3318:
3317:
3312:
3309:
3304:
3301:
3300:
3294:
3291:on 2011-07-20
3287:
3280:
3279:
3274:
3273:Babai, László
3270:
3267:on 2007-07-14
3266:
3262:
3261:
3256:
3252:
3250:
3247:
3246:
3242:
3232:
3231:
3226:
3225:Spencer, Joel
3222:
3218:
3217:Aldous, David
3214:
3210:
3206:
3202:
3198:
3194:
3189:
3184:
3180:
3176:
3175:
3170:
3166:
3162:
3158:
3154:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3099:
3094:
3090:
3085:
3080:
3076:
3072:
3071:
3066:
3062:
3058:
3054:
3050:
3048:0-387-24219-8
3044:
3040:
3039:
3033:
3028:
3023:
3019:
3015:
2999:
2991:
2987:
2983:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2954:
2949:
2945:
2941:
2937:
2933:
2929:
2925:
2921:
2920:
2915:
2911:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2878:
2873:
2868:
2864:
2860:
2856:
2852:
2848:
2844:
2843:
2838:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2808:
2803:
2799:
2796:
2790:
2786:
2782:
2778:
2777:Berge, Claude
2774:
2773:
2769:
2761:
2756:
2753:
2749:
2744:
2741:
2737:
2732:
2729:
2725:
2720:
2717:
2713:
2712:Harzheim 2005
2708:
2706:
2702:
2698:
2693:
2690:
2686:
2681:
2678:
2674:
2673:Dilworth 1950
2669:
2666:
2659:
2657:
2654:
2649:
2647:
2643:
2638:
2636:
2632:
2628:
2624:
2620:
2616:
2611:
2609:
2605:
2601:
2582:
2566:
2562:
2558:
2550:
2539:
2531:
2527:
2520:
2517:
2510:
2509:
2508:
2505:
2501:
2497:
2493:
2490:}—ordered by
2489:
2485:
2482:-element set
2481:
2477:
2472:
2468:
2465:
2457:
2455:
2453:
2449:
2445:
2444:Lovász (1972)
2441:
2437:
2433:
2428:
2426:
2422:
2418:
2414:
2410:
2402:
2400:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2370:
2366:
2360:
2350:
2348:
2347:Perles (1963)
2344:
2342:
2335:
2329:
2327:
2323:
2319:
2315:
2311:
2307:
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2275:
2271:
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2263:
2259:
2255:
2251:
2247:
2239:
2237:
2235:
2231:
2227:
2223:
2219:
2214:
2212:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2165:exactly when
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2135:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2082:has at least
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1972:
1970:
1966:
1962:
1953:
1944:
1928:
1924:
1920:
1917:
1914:
1909:
1905:
1901:
1896:
1892:
1888:
1882:
1859:
1856:
1853:
1833:
1813:
1793:
1773:
1753:
1750:
1747:
1724:
1718:
1715:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1668:
1665:
1643:
1639:
1635:
1632:
1612:
1592:
1572:
1566:
1546:
1543:
1540:
1515:
1511:
1501:
1481:
1478:
1475:
1455:
1449:
1429:
1409:
1403:
1381:
1377:
1351:
1347:
1343:
1340:
1337:
1332:
1328:
1324:
1321:
1315:
1309:
1299:be the chain
1286:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1239:
1236:
1214:
1210:
1206:
1203:
1183:
1174:
1160:
1138:
1134:
1130:
1125:
1121:
1100:
1080:
1060:
1057:
1052:
1048:
1025:
1021:
1017:
1012:
1008:
985:
981:
958:
954:
950:
947:
925:
921:
917:
912:
908:
904:
901:
878:
873:
869:
865:
860:
856:
835:
815:
793:
789:
768:
746:
742:
721:
696:
692:
688:
685:
682:
677:
673:
669:
664:
660:
653:
650:
629:
626:
605:
583:
579:
556:
552:
531:
528:
525:
522:
519:
516:
513:
510:
507:
487:
484:
481:
478:
475:
472:
469:
466:
463:
440:
435:
431:
427:
422:
418:
397:
375:
371:
348:
344:
340:
337:
334:
329:
325:
304:
281:
272:
269:
265:
262:
241:
232:
218:
198:
178:
158:
138:
129:
127:
123:
107:
95:
93:
90:
85:
77:
75:
71:
69:
65:
61:
57:
53:
49:
45:
44:combinatorics
41:
37:
30:
19:
4031:Order theory
3802:& Orders
3780:Star product
3709:Well-founded
3662:Prefix order
3618:Distributive
3608:Complemented
3578:Foundational
3543:Completeness
3499:Zorn's lemma
3458:
3403:Cyclic order
3386:Key concepts
3356:Order theory
3314:
3298:
3286:the original
3277:
3265:the original
3258:
3229:
3178:
3172:
3134:
3102:
3096:
3093:Mirsky, Leon
3074:
3068:
3038:Ordered sets
3037:
3020:(1): 41–68,
3017:
3013:
3012:-families",
2957:
2951:
2948:Galvin, Fred
2923:
2917:
2881:
2875:
2849:(1): 23–32,
2846:
2840:
2811:
2805:
2784:
2755:
2743:
2731:
2719:
2692:
2680:
2668:
2650:
2639:
2634:
2630:
2626:
2622:
2618:
2615:divisibility
2612:
2603:
2599:
2597:
2503:
2499:
2487:
2483:
2479:
2470:
2466:
2461:
2429:
2406:
2396:
2392:
2388:
2384:
2380:
2376:
2372:
2368:
2362:
2345:
2330:
2325:
2321:
2313:
2309:
2305:
2301:
2297:
2293:
2289:
2285:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2243:
2233:
2229:
2225:
2221:
2215:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2136:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1973:
1958:
1175:
233:
130:
99:
81:
72:
63:
47:
40:order theory
33:
3986:Riesz space
3947:Isomorphism
3823:Normal cone
3745:Composition
3679:Semilattice
3588:Homogeneous
3573:Equivalence
3423:Total order
2748:Steele 1995
2724:Mirsky 1971
2653:antimatroid
2010:and where (
410:. Clearly,
36:mathematics
4025:Categories
3954:Order type
3888:Cofinality
3729:Well-order
3704:Transitive
3593:Idempotent
3526:Asymmetric
3260:PlanetMath
2770:References
2448:complement
2300:has width
2256:has width
643:, and set
4005:Upper set
3942:Embedding
3878:Antichain
3699:Tolerance
3689:Symmetric
3684:Semiorder
3630:Reflexive
3548:Connected
3316:MathWorld
3205:120943065
2863:119826035
2571:⌋
2555:⌊
2521:
2492:inclusion
2476:power set
2157:in which
1918:…
1719:∪
1687:…
1669:∈
1658:for each
1585:. Thus,
1570:∖
1544:−
1505:∖
1479:−
1453:∖
1407:∖
1344:≤
1325:∈
1316:∪
1258:…
1240:∈
1229:for some
1207:≥
951:≤
918:∩
905:∈
882:∅
879:≠
866:∩
686:…
526:…
482:…
444:∅
441:≠
428:∩
338:…
276:∖
84:antichain
78:Statement
56:antichain
3800:Topology
3667:Preorder
3650:Eulerian
3613:Complete
3563:Directed
3553:Covering
3418:Preorder
3377:Category
3372:Glossary
3275:(2005),
2783:(1984),
2278:coloring
1998:) where
1636:≱
1131:≱
1058:≱
1040:, since
1018:≱
630:′
390:of size
266:′
89:disjoint
3905:Duality
3883:Cofinal
3871:Related
3850:Fréchet
3727:)
3603:Bounded
3598:Lattice
3571:)
3569:Partial
3437:Results
3408:Lattice
3197:0168497
3161:2920058
3119:2316481
3057:2127991
2982:1270960
2974:2975628
2940:2033375
2906:1363140
2898:2079151
2828:1969503
2474:is the
2432:perfect
2280:of the
2122:; then
2078:; then
2066:. Let
1872:chains
1826:). Now
1786:(since
1708:, then
940:. Then
848:. Then
124: (
3930:Subnet
3910:Filter
3860:Normed
3845:Banach
3811:&
3718:Better
3655:Strict
3645:Graded
3536:topics
3367:Topics
3203:
3195:
3159:
3149:
3117:
3055:
3045:
2980:
2972:
2938:
2904:
2896:
2861:
2826:
2791:
2478:of an
2446:, the
2417:clique
2411:is an
2312:has a
2177:is in
2169:is in
1279:. Let
894:. Let
544:, let
500:. For
122:Galvin
60:chains
3920:Ideal
3898:Graph
3694:Total
3672:Total
3558:Dense
3289:(PDF)
3282:(PDF)
3234:(PDF)
3201:S2CID
3115:JSTOR
2970:JSTOR
2936:JSTOR
2902:S2CID
2877:Order
2859:S2CID
2842:Order
2824:JSTOR
2660:Notes
2648:two.
2518:width
2185:from
2161:<
2118:) in
2026:<
2022:when
1978:with
64:width
3511:list
3147:ISBN
3043:ISBN
2789:ISBN
2640:The
2462:The
2236:).
2126:has
2106:and
2058:and
1093:and
828:and
456:for
131:Let
126:1994
42:and
3925:Net
3725:Pre
3183:doi
3139:doi
3107:doi
3079:doi
3022:doi
2962:doi
2958:101
2928:doi
2886:doi
2851:doi
2816:doi
2442:of
2308:of
2284:of
2268:of
2189:to
2141:= (
2046:in
2038:in
2030:in
1986:= (
1963:on
1766:in
1559:in
618:in
128:).
82:An
34:In
4027::
3313:.
3257:,
3223:;
3219:;
3199:,
3193:MR
3191:,
3177:,
3157:MR
3155:,
3145:,
3129:;
3113:,
3103:78
3101:,
3073:,
3053:MR
3051:,
3018:20
3016:,
2978:MR
2976:,
2968:,
2956:,
2934:,
2922:,
2900:,
2894:MR
2892:,
2882:20
2880:,
2874:,
2857:,
2845:,
2822:,
2812:51
2810:,
2779:;
2704:^
2637:.
2407:A
2320:,
2234:kn
2173:,
2130:-
2086:-
2006:=
2002:=
1971:.
1396:,
654::=
270::=
231:.
46:,
3723:(
3720:)
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3567:(
3514:)
3348:e
3341:t
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3319:.
3238:.
3208:.
3185::
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3164:.
3141::
3122:.
3109::
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3081::
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3060:.
3031:.
3024::
3000:k
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2964::
2943:.
2930::
2924:7
2909:.
2888::
2866:.
2853::
2847:5
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2818::
2762:.
2750:.
2738:.
2726:.
2714:.
2699:.
2687:.
2675:.
2635:n
2631:m
2627:m
2623:n
2619:n
2604:X
2600:X
2583:.
2577:)
2567:2
2563:/
2559:n
2551:n
2546:(
2540:=
2537:)
2532:n
2528:B
2524:(
2504:n
2500:B
2488:n
2484:X
2480:n
2471:n
2467:B
2397:N
2393:i
2391:(
2389:N
2385:x
2381:x
2379:(
2377:N
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2369:x
2340:0
2338:ℵ
2326:w
2322:P
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2246:w
2232:(
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2187:u
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2151:E
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2100:P
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2084:n
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2068:A
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2036:M
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2016:v
2014:,
2012:u
2008:S
2004:V
2000:U
1996:E
1994:,
1992:V
1990:,
1988:U
1984:G
1980:n
1976:S
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