1180:
844:
881:
variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same
868:
Note that it is important that all sets are closed, i.e., contain their boundary. If, for example, the red set is not closed, then it is possible that the central point is contained only in the blue and green sets, and then the intersection of all three sets may be empty.
2484:. Therefore, there exists a balanced collection such that the corresponding sets in the new covering have nonempty intersection. But the only possible balanced collection is the collection of all singletons; hence, the original covering has nonempty intersection.
1313:
faces. However, there are connector-free coverings that are not KKM coverings. An example is illustrated at the right. There, the red set touches all three faces, but it does not contain any connector, since no connected component of it touches all three faces.
1156:"A colloquial statement of this result is... if each of three people paint a triangle red, white and blue according to the KKM rules, then there will be a point which is in the red set of one person, the white set of another, the blue of the third".
1653:
1055:
2970:
2052:
The collection {{1}, {2}, {3}} is balanced: choose all weights to be 1. The same is true for any collection in which each element appears exactly once, such as the collection {{1,2},{3}} or the collection { {1,2,3}
2738:
1146:
3101:
2059:
The collection {{1,2}, {2,3}} is not balanced, since for any choice of positive weights, the sum for element 2 will be larger than the sum for element 1 or 3, so it is not possible that all sums equal 1.
415:
2856:
1836:
2621:
3009:
2784:
2143:
273:
2660:
2421:
344:
2551:
1250:
225:
3300:
2311:
838:
529:
3155:
3246:
3037:
2235:
1484:
97:
2482:
2265:
1942:
1516:
476:
172:
2343:
1701:
1422:
1968:
3213:
3182:
2452:
2197:
2000:
1307:
1277:
778:
751:
724:
697:
670:
643:
613:
586:
559:
129:
1762:
1572:
1075:
3310:
Oleg R. Musin proved several generalizations of the KKM lemma and KKMS theorem, with boundary conditions on the coverings. The boundary conditions are related to
2046:
1349:
449:
2879:
2804:
2383:
2363:
2056:
The collection {{1,2}, {2,3}, {3,1}} is balanced: choose all weights to be 1/2. The same is true for any collection in which each element appears exactly twice.
2020:
1909:
1859:
1782:
1729:
1564:
1540:
297:
1885:
1451:
851:
The lemma is illustrated by the picture on the right, in which set #1 is blue, set #2 is red and set #3 is green. The KKM requirements are satisfied, since:
3324:
956:
2888:
2665:
3983:
3758:
Zhou, Lin (1994). "A Theorem on Open
Coverings of a Simplex and Scarf's Core Existence Theorem through Brouwer's Fixed Point Theorem".
926:
865:
The KKM lemma states that there is a point covered by all three colors simultaneously; such a point is clearly visible in the picture.
1083:
3534:
3405:
3042:
2861:
Komiya's theorem also generalizes the definition of a balanced collection: instead of requiring that there is a weight function on
3550:
Shapley, Lloyd; Vohra, Rajiv (1991). "On
Kakutani's fixed point theorem, the K-K-M-S theorem and the core of a balanced game".
1703:
sets in a KKMS covering is nonempty; this is illustrated by the special case of a KKM covering, in which most sets are empty.
1663:
sets corresponding to singletons are nonempty, while the other sets are empty. However, there are many other KKMS coverings.
363:
3623:
Krasa, Stefan; Yannelis, Nicholas C. (1994). "An elementary proof of the
Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem".
2812:
1792:
3978:
1354:
The connector-free variant also has a permutation variant, so that both these generalizations can be used simultaneously.
2568:
899:
55:
2975:
2750:
2065:
234:
2626:
916:
3366:
2150:
1377:
306:
2523:
1209:
949:
proved the following generalization of the KKM lemma. Suppose that, instead of one KKM covering, we have
184:
3251:
2390:
790:
481:
2509:
3349:
3345:
3121:
40:
36:
3426:
3353:
3218:
3018:
2202:
1456:
69:
44:
32:
2273:
3940:
3914:
3881:
3863:
3836:
3818:
3791:
3783:
3713:
3678:
3640:
3567:
3505:
3467:
3422:
3108:
1914:
1489:
878:
454:
145:
1322:
1161:
904:
51:
478:
which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets
3932:
3775:
3530:
921:
139:
2316:
1673:
1648:{\displaystyle \operatorname {conv} (\{v_{i}:i\in I\})\subseteq \bigcup _{J\subseteq I}C_{J}}
1394:
3924:
3873:
3828:
3767:
3740:
3705:
3670:
3632:
3603:
3559:
3497:
3459:
3414:
3375:
2457:
2240:
1947:
1543:
300:
3434:
3191:
3160:
2430:
2175:
1973:
1285:
1255:
756:
729:
702:
675:
648:
621:
591:
564:
537:
102:
3988:
3430:
1734:
1060:
3488:(1989). "A constructive proof of a permutation-based generalization of Sperner's lemma".
3418:
2025:
1328:
428:
1179:
3485:
2864:
2789:
2368:
2348:
2005:
1894:
1844:
1767:
1714:
1549:
1525:
1325:, implies the KKM lemma extends to connector-free coverings (he proved his theorem for
1318:
282:
3744:
3400:
1864:
1430:
3972:
3944:
3905:
3885:
3840:
3795:
3717:
3696:
Herings, P. Jean-Jacques (1997). "An extremely simple proof of the K-K-M-S Theorem".
3682:
3608:
3591:
3571:
3471:
1369:
1192:
3644:
3509:
877:
There are several fixed-point theorems which come in three equivalent variants: an
3731:
Reny, Philip J.; Holtz
Wooders, Myrna (1998). "An extension of the KKMS theorem".
3524:
1519:
276:
228:
1050:{\displaystyle C_{1}^{1},\ldots ,C_{n}^{1},\ldots ,C_{1}^{n},\ldots ,C_{n}^{n}}
3928:
3901:"Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals"
3832:
3396:
3185:
1659:
Any KKM covering is a special case of a KKMS covering. In a KKM covering, the
1424:
946:
3936:
3877:
3779:
3118:
The KKMS theorem is a special case of Komiya's theorem in which the polytope
17:
3900:
3854:
Musin, Oleg R. (2016). "Homotopy invariants of covers and KKM type lemmas".
3380:
2965:{\displaystyle {\textbf {b}}=\{b^{F}:F\in {\textrm {Faces}}(P),b^{F}\in F\}}
1373:
1167:
The original KKM lemma follows from the rainbow KKM lemma by simply picking
1152:
The name "rainbow KKM lemma" is inspired by Gale's description of his lemma:
1282:
Any KKM covering is a connector-free covering, since in a KKM covering, no
3709:
3450:
Gale, D. (1984). "Equilibrium in a discrete exchange economy with money".
3311:
2513:
2498:
2497:
Zhou proved a variant of the KKMS theorem where the covering consists of
3787:
843:
3771:
3674:
3636:
3563:
3501:
3463:
2490:
Reny and
Wooders proved that the balanced set can also be chosen to be
2172:
The KKMS theorem implies the KKM lemma. Suppose we have a KKM covering
132:
3809:
Musin, Oleg R. (2017). "KKM type theorems with boundary conditions".
847:
An example of a covering satisfying the requirements of the KKM lemma
1841:
It remains to explain what a "balanced collection" is. A collection
3919:
3868:
3823:
2733:
F\subseteq \bigcup _{G\subseteq F,~G\in {\textrm {Faces}}(P)}C_{G}.
3962:
1178:
842:
3401:"A Borsuk–Ulam equivalent that directly implies Sperner's lemma"
1141:{\displaystyle \bigcap _{i=1}^{n}C_{i}^{\pi (i)}\neq \emptyset }
3661:
Komiya, Hidetoshi (1994). "A simple proof of K-K-M-S theorem".
3357:
618:
The edge 12 (from vertex 1 to vertex 2) is covered by the sets
3096:{\displaystyle b^{P}\in \operatorname {conv} \{b^{F}:F\in B\}}
858:
Each edge is covered by the two colors of its two vertices.
3592:"On the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem"
2740:
Komiya's theorem says that for every Komiya covering of
2062:
The collection {{1,2}, {2,3}, {1}} is balanced: choose
410:{\displaystyle \bigcap _{i=1}^{n}C_{i}\neq \emptyset .}
3111:
of the points assigned to the faces in the collection
783:
The union of all three sets covers the entire triangle
351:
in every KKM covering, the common intersection of all
3254:
3221:
3194:
3163:
3124:
3045:
3021:
2978:
2891:
2867:
2851:{\displaystyle \bigcap _{F\in B}C_{F}\neq \emptyset }
2815:
2792:
2753:
2629:
2571:
2526:
2460:
2433:
2393:
2371:
2351:
2319:
2276:
2243:
2205:
2178:
2068:
2028:
2008:
1976:
1950:
1917:
1897:
1867:
1847:
1831:{\displaystyle \bigcap _{J\in B}C_{J}\neq \emptyset }
1795:
1770:
1737:
1717:
1676:
1575:
1552:
1528:
1492:
1459:
1433:
1397:
1331:
1288:
1258:
1212:
1183:
An illustration of the generalized KKM lemma by Bapat
1086:
1077:
of the coverings with a non-empty intersection, i.e:
1063:
959:
793:
759:
732:
705:
678:
651:
624:
594:
567:
540:
484:
457:
431:
366:
309:
285:
237:
187:
148:
105:
72:
3103:, that is, the point assigned to the entire polygon
2616:{\displaystyle \{C_{F}:F\in {\textrm {Faces}}(P)\}}
3294:
3240:
3207:
3176:
3149:
3095:
3031:
3003:
2964:
2873:
2850:
2798:
2778:
2732:
2654:
2615:
2545:
2476:
2446:
2415:
2377:
2357:
2337:
2305:
2259:
2229:
2191:
2137:
2040:
2014:
1994:
1962:
1936:
1903:
1879:
1853:
1830:
1776:
1756:
1723:
1695:
1647:
1558:
1534:
1510:
1478:
1445:
1416:
1343:
1301:
1271:
1244:
1140:
1069:
1049:
832:
772:
745:
718:
691:
664:
637:
607:
580:
553:
523:
470:
443:
409:
338:
291:
267:
219:
166:
123:
91:
3596:Journal of Mathematical Analysis and Applications
2881:such that the sum of weights near each vertex of
1546:by the union of sets corresponding to subsets of
3323:A common generalization of the KKMS theorem and
2786:, such that the intersection of sets indexed by
1764:, such that the intersection of sets indexed by
3004:{\displaystyle B\subseteq {\textrm {Faces}}(P)}
2779:{\displaystyle B\subseteq {\textrm {Faces}}(P)}
2512:generalized the KKMS theorem from simplices to
2454:implies the KKMS condition on the new covering
2002:, the sum of weights of all subsets containing
3811:Journal of Fixed Point Theory and Applications
2885:is 1, we start by choosing any set of points
8:
3899:Frick, Florian; Zerbib, Shira (2019-06-01).
3090:
3065:
2959:
2902:
2610:
2572:
2332:
2326:
1610:
1585:
1160:The rainbow KKM lemma can be proved using a
861:The triangle is covered by all three colors.
262:
244:
2427:The KKM condition on the original covering
2157:is balanced with respect to its ground-set
2138:{\displaystyle w_{1,2}=0,w_{2,3}=1,w_{1}=1}
2365:is a singleton that contains only element
268:{\displaystyle I\subseteq \{1,\ldots ,n\}}
3918:
3867:
3822:
3607:
3379:
3281:
3261:
3253:
3226:
3220:
3199:
3193:
3168:
3162:
3135:
3123:
3072:
3050:
3044:
3023:
3022:
3020:
2986:
2985:
2977:
2947:
2925:
2924:
2909:
2893:
2892:
2890:
2866:
2836:
2820:
2814:
2791:
2761:
2760:
2752:
2722:
2701:
2700:
2678:
2655:{\displaystyle F\in {\textrm {Faces}}(P)}
2637:
2636:
2628:
2595:
2594:
2579:
2570:
2528:
2527:
2525:
2465:
2459:
2438:
2432:
2398:
2392:
2370:
2350:
2318:
2297:
2281:
2275:
2248:
2242:
2204:
2183:
2177:
2123:
2098:
2073:
2067:
2027:
2007:
1975:
1949:
1922:
1916:
1896:
1866:
1846:
1816:
1800:
1794:
1769:
1742:
1736:
1716:
1681:
1675:
1670:true that the common intersection of all
1639:
1623:
1592:
1574:
1551:
1527:
1491:
1464:
1458:
1432:
1402:
1396:
1330:
1293:
1287:
1263:
1257:
1236:
1217:
1211:
1162:rainbow generalization of Sperner's lemma
1117:
1112:
1102:
1091:
1085:
1062:
1041:
1036:
1017:
1012:
993:
988:
969:
964:
958:
855:Each vertex is covered by a unique color.
824:
811:
798:
792:
764:
758:
737:
731:
710:
704:
683:
677:
656:
650:
629:
623:
599:
593:
572:
566:
545:
539:
515:
502:
489:
483:
462:
456:
430:
392:
382:
371:
365:
330:
314:
308:
284:
236:
211:
192:
186:
147:
104:
77:
71:
1368:is a generalization of the KKM lemma by
884:
3337:
3526:Modeling, Computation and Optimization
2169:admits a perfect fractional matching.
451:, the KKM lemma considers the simplex
339:{\displaystyle \bigcup _{i\in I}C_{i}}
3585:
3583:
3581:
3445:
3443:
2487:The KKMS theorem has various proofs.
2161:, iff the hypergraph with vertex-set
1427:- indexed by the nonempty subsets of
910:Knaster–Kuratowski–Mazurkiewicz lemma
726:, the edge 31 is covered by the sets
672:, the edge 23 is covered by the sets
29:Knaster–Kuratowski–Mazurkiewicz lemma
7:
3656:
3654:
3452:International Journal of Game Theory
3427:10.4169/amer.math.monthly.120.04.346
3419:10.4169/amer.math.monthly.120.04.346
2546:{\displaystyle {\textrm {Faces}}(P)}
2520:be any compact convex polytope. Let
1453:(equivalently: by nonempty faces of
3358:"Ein Beweis des Fixpunktsatzes für
3024:
2894:
2022:is exactly 1. For example, suppose
1708:in every KKMS covering, there is a
1245:{\displaystyle C_{1},\ldots ,C_{n}}
1057:. Then, there exists a permutation
840:have at least one point in common.
787:The KKM lemma states that the sets
220:{\displaystyle C_{1},\ldots ,C_{n}}
3856:Algebraic & Geometric Topology
3223:
3132:
2845:
2410:
1825:
1461:
1135:
459:
401:
74:
31:is a basic result in mathematical
25:
3733:Journal of Mathematical Economics
3406:The American Mathematical Monthly
3295:{\displaystyle (1/n,\ldots ,1/n)}
2416:{\displaystyle C'_{J}=\emptyset }
1891:if there is a weight function on
1522:of the vertices corresponding to
833:{\displaystyle C_{1},C_{2},C_{3}}
524:{\displaystyle C_{1},C_{2},C_{3}}
279:of the vertices corresponding to
50:The KKM lemma can be proved from
2553:be the set of nonempty faces of
3150:{\displaystyle P=\Delta _{n-1}}
2505:Polytopal KKMS theorem (Komiya)
1970:), such that, for each element
3961:See the proof of KKM Lemma in
3523:Bapat, Ravindra (2009-04-03).
3289:
3255:
3188:of the face F (in particular,
2998:
2992:
2937:
2931:
2773:
2767:
2713:
2707:
2649:
2643:
2607:
2601:
2540:
2534:
1989:
1983:
1874:
1868:
1749:
1743:
1613:
1582:
1505:
1499:
1440:
1434:
1383:While a KKM covering contains
1127:
1121:
927:Lusternik–Schnirelmann theorem
118:
106:
1:
3745:10.1016/S0304-4068(97)00004-9
3241:{\displaystyle \Delta _{n-1}}
3032:{\displaystyle {\textbf {b}}}
2230:{\displaystyle i=1,\ldots ,n}
1479:{\displaystyle \Delta _{n-1}}
92:{\displaystyle \Delta _{n-1}}
54:and can be used to prove the
3609:10.1016/0022-247X(81)90063-9
2306:{\displaystyle C'_{J}=C_{i}}
2237:. Construct a KKMS covering
1706:The KKMS theorem says that,
1175:Connector-free lemma (Bapat)
3306:Boundary conditions (Musin)
2565:is a family of closed sets
1937:{\displaystyle w_{J}\geq 0}
1511:{\displaystyle I\subseteq }
900:Brouwer fixed-point theorem
471:{\displaystyle \Delta _{2}}
167:{\displaystyle 1,\ldots ,n}
56:Brouwer fixed-point theorem
4005:
3984:Fixed points (mathematics)
3590:Ichiishi, Tatsuro (1981).
3929:10.1007/s00493-018-3891-1
3833:10.1007/s11784-016-0388-7
2623:such that for every face
2501:rather than closed sets.
953:different KKM coverings:
3878:10.2140/agt.2016.16.1799
3490:Mathematical Programming
941:Rainbow KKM lemma (Gale)
349:The KKM lemma says that
3381:10.4064/fm-14-1-132-137
3367:Fundamenta Mathematicae
3362:-dimensionale Simplexe"
2338:{\displaystyle J=\{i\}}
1696:{\displaystyle 2^{n}-1}
1417:{\displaystyle 2^{n}-1}
1378:cooperative game theory
1204:connector-free covering
3325:Carathéodory's theorem
3296:
3242:
3209:
3178:
3151:
3097:
3033:
3005:
2966:
2875:
2852:
2800:
2780:
2734:
2656:
2617:
2547:
2478:
2477:{\displaystyle C'_{J}}
2448:
2417:
2379:
2359:
2339:
2307:
2261:
2260:{\displaystyle C'_{J}}
2231:
2193:
2151:hypergraph terminology
2139:
2042:
2016:
1996:
1964:
1963:{\displaystyle J\in B}
1938:
1905:
1881:
1855:
1832:
1778:
1758:
1725:
1697:
1657:
1649:
1560:
1536:
1512:
1480:
1447:
1418:
1345:
1303:
1279:contains a connector.
1273:
1246:
1199:faces of the simplex.
1184:
1158:
1142:
1107:
1071:
1051:
848:
834:
774:
747:
720:
693:
666:
639:
609:
582:
555:
525:
472:
445:
411:
387:
340:
293:
269:
221:
168:
125:
93:
3710:10.1007/s001990050161
3297:
3248:, which is the point
3243:
3215:is the barycenter of
3210:
3208:{\displaystyle b^{P}}
3179:
3177:{\displaystyle b^{F}}
3152:
3098:
3034:
3006:
2967:
2876:
2853:
2801:
2781:
2735:
2657:
2618:
2548:
2479:
2449:
2447:{\displaystyle C_{i}}
2418:
2380:
2360:
2340:
2308:
2262:
2232:
2194:
2192:{\displaystyle C_{i}}
2140:
2043:
2017:
1997:
1995:{\displaystyle i\in }
1965:
1939:
1906:
1882:
1856:
1833:
1779:
1759:
1726:
1698:
1650:
1568:
1561:
1537:
1513:
1481:
1448:
1419:
1346:
1304:
1302:{\displaystyle C_{i}}
1274:
1272:{\displaystyle C_{i}}
1247:
1182:
1171:identical coverings.
1154:
1143:
1087:
1072:
1052:
846:
835:
775:
773:{\displaystyle C_{1}}
748:
746:{\displaystyle C_{3}}
721:
719:{\displaystyle C_{3}}
694:
692:{\displaystyle C_{2}}
667:
665:{\displaystyle C_{2}}
640:
638:{\displaystyle C_{1}}
610:
608:{\displaystyle C_{3}}
583:
581:{\displaystyle C_{2}}
556:
554:{\displaystyle C_{1}}
526:
473:
446:
412:
367:
341:
294:
270:
222:
169:
126:
124:{\displaystyle (n-1)}
94:
35:published in 1929 by
3979:Fixed-point theorems
3529:. World Scientific.
3252:
3219:
3192:
3161:
3122:
3043:
3019:
2976:
2889:
2865:
2813:
2790:
2751:
2666:
2627:
2569:
2524:
2458:
2431:
2391:
2369:
2349:
2317:
2274:
2241:
2203:
2176:
2066:
2026:
2006:
1974:
1948:
1915:
1911:(assigning a weight
1895:
1865:
1845:
1793:
1768:
1757:{\displaystyle 2^{}}
1735:
1715:
1674:
1573:
1550:
1526:
1490:
1457:
1431:
1395:
1329:
1286:
1256:
1210:
1084:
1070:{\displaystyle \pi }
1061:
957:
791:
757:
730:
703:
676:
649:
622:
592:
565:
538:
482:
455:
429:
364:
307:
283:
235:
185:
181:is defined as a set
146:
103:
70:
3395:Nyman, Kathryn L.;
2746:balanced collection
2473:
2406:
2289:
2256:
2041:{\displaystyle n=3}
1710:balanced collection
1344:{\displaystyle n=3}
1131:
1046:
1022:
998:
974:
917:Borsuk–Ulam theorem
444:{\displaystyle n=3}
3772:10.1007/BF01215385
3675:10.1007/BF01215383
3637:10.1007/BF01215384
3564:10.1007/BF01210576
3502:10.1007/BF01587081
3464:10.1007/BF01769865
3397:Su, Francis Edward
3292:
3238:
3205:
3174:
3147:
3109:convex combination
3093:
3029:
3001:
2962:
2871:
2848:
2831:
2796:
2776:
2717:
2652:
2613:
2543:
2474:
2461:
2444:
2413:
2394:
2375:
2355:
2335:
2303:
2277:
2257:
2244:
2227:
2189:
2135:
2038:
2012:
1992:
1960:
1934:
1901:
1877:
1851:
1828:
1811:
1774:
1754:
1721:
1693:
1666:in general, it is
1645:
1634:
1556:
1532:
1508:
1476:
1443:
1414:
1372:. It is useful in
1341:
1299:
1269:
1242:
1191:of a simplex is a
1185:
1138:
1108:
1067:
1047:
1032:
1008:
984:
960:
888:Algebraic topology
879:algebraic topology
873:Equivalent results
849:
830:
770:
743:
716:
689:
662:
635:
605:
578:
551:
521:
468:
441:
407:
336:
325:
289:
265:
231:such that for any
217:
164:
121:
89:
33:fixed-point theory
3026:
2989:
2928:
2896:
2874:{\displaystyle B}
2816:
2799:{\displaystyle B}
2764:
2704:
2693:
2674:
2640:
2598:
2531:
2378:{\displaystyle i}
2358:{\displaystyle J}
2015:{\displaystyle i}
1904:{\displaystyle B}
1854:{\displaystyle B}
1796:
1777:{\displaystyle B}
1724:{\displaystyle B}
1619:
1559:{\displaystyle I}
1535:{\displaystyle I}
1309:even touches all
1195:that touches all
932:
931:
588:covers vertex 2,
561:covers vertex 1,
310:
292:{\displaystyle I}
16:(Redirected from
3996:
3949:
3948:
3922:
3896:
3890:
3889:
3871:
3862:(3): 1799–1812.
3851:
3845:
3844:
3826:
3817:(3): 2037–2049.
3806:
3800:
3799:
3755:
3749:
3748:
3728:
3722:
3721:
3693:
3687:
3686:
3658:
3649:
3648:
3620:
3614:
3613:
3611:
3587:
3576:
3575:
3547:
3541:
3540:
3520:
3514:
3513:
3496:(1–3): 113–120.
3482:
3476:
3475:
3447:
3438:
3437:
3392:
3386:
3384:
3383:
3354:Mazurkiewicz, S.
3342:
3301:
3299:
3298:
3293:
3285:
3265:
3247:
3245:
3244:
3239:
3237:
3236:
3214:
3212:
3211:
3206:
3204:
3203:
3183:
3181:
3180:
3175:
3173:
3172:
3156:
3154:
3153:
3148:
3146:
3145:
3102:
3100:
3099:
3094:
3077:
3076:
3055:
3054:
3038:
3036:
3035:
3030:
3028:
3027:
3015:with respect to
3010:
3008:
3007:
3002:
2991:
2990:
2987:
2971:
2969:
2968:
2963:
2952:
2951:
2930:
2929:
2926:
2914:
2913:
2898:
2897:
2880:
2878:
2877:
2872:
2857:
2855:
2854:
2849:
2841:
2840:
2830:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2766:
2765:
2762:
2739:
2737:
2736:
2731:
2727:
2726:
2716:
2706:
2705:
2702:
2691:
2661:
2659:
2658:
2653:
2642:
2641:
2638:
2622:
2620:
2619:
2614:
2600:
2599:
2596:
2584:
2583:
2552:
2550:
2549:
2544:
2533:
2532:
2529:
2510:Hidetoshi Komiya
2483:
2481:
2480:
2475:
2469:
2453:
2451:
2450:
2445:
2443:
2442:
2422:
2420:
2419:
2414:
2402:
2384:
2382:
2381:
2376:
2364:
2362:
2361:
2356:
2344:
2342:
2341:
2336:
2312:
2310:
2309:
2304:
2302:
2301:
2285:
2266:
2264:
2263:
2258:
2252:
2236:
2234:
2233:
2228:
2198:
2196:
2195:
2190:
2188:
2187:
2144:
2142:
2141:
2136:
2128:
2127:
2109:
2108:
2084:
2083:
2047:
2045:
2044:
2039:
2021:
2019:
2018:
2013:
2001:
1999:
1998:
1993:
1969:
1967:
1966:
1961:
1943:
1941:
1940:
1935:
1927:
1926:
1910:
1908:
1907:
1902:
1886:
1884:
1883:
1880:{\displaystyle }
1878:
1860:
1858:
1857:
1852:
1837:
1835:
1834:
1829:
1821:
1820:
1810:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1753:
1752:
1730:
1728:
1727:
1722:
1702:
1700:
1699:
1694:
1686:
1685:
1654:
1652:
1651:
1646:
1644:
1643:
1633:
1597:
1596:
1565:
1563:
1562:
1557:
1541:
1539:
1538:
1533:
1517:
1515:
1514:
1509:
1485:
1483:
1482:
1477:
1475:
1474:
1452:
1450:
1449:
1446:{\displaystyle }
1444:
1423:
1421:
1420:
1415:
1407:
1406:
1376:, especially in
1360:
1350:
1348:
1347:
1342:
1308:
1306:
1305:
1300:
1298:
1297:
1278:
1276:
1275:
1270:
1268:
1267:
1251:
1249:
1248:
1243:
1241:
1240:
1222:
1221:
1147:
1145:
1144:
1139:
1130:
1116:
1106:
1101:
1076:
1074:
1073:
1068:
1056:
1054:
1053:
1048:
1045:
1040:
1021:
1016:
997:
992:
973:
968:
942:
885:
839:
837:
836:
831:
829:
828:
816:
815:
803:
802:
779:
777:
776:
771:
769:
768:
752:
750:
749:
744:
742:
741:
725:
723:
722:
717:
715:
714:
698:
696:
695:
690:
688:
687:
671:
669:
668:
663:
661:
660:
644:
642:
641:
636:
634:
633:
615:covers vertex 3.
614:
612:
611:
606:
604:
603:
587:
585:
584:
579:
577:
576:
560:
558:
557:
552:
550:
549:
530:
528:
527:
522:
520:
519:
507:
506:
494:
493:
477:
475:
474:
469:
467:
466:
450:
448:
447:
442:
416:
414:
413:
408:
397:
396:
386:
381:
355:sets is nonempty
345:
343:
342:
337:
335:
334:
324:
298:
296:
295:
290:
274:
272:
271:
266:
226:
224:
223:
218:
216:
215:
197:
196:
173:
171:
170:
165:
130:
128:
127:
122:
98:
96:
95:
90:
88:
87:
21:
4004:
4003:
3999:
3998:
3997:
3995:
3994:
3993:
3969:
3968:
3958:
3953:
3952:
3898:
3897:
3893:
3853:
3852:
3848:
3808:
3807:
3803:
3760:Economic Theory
3757:
3756:
3752:
3730:
3729:
3725:
3698:Economic Theory
3695:
3694:
3690:
3663:Economic Theory
3660:
3659:
3652:
3625:Economic Theory
3622:
3621:
3617:
3589:
3588:
3579:
3552:Economic Theory
3549:
3548:
3544:
3537:
3522:
3521:
3517:
3484:
3483:
3479:
3449:
3448:
3441:
3394:
3393:
3389:
3344:
3343:
3339:
3334:
3320:
3308:
3250:
3249:
3222:
3217:
3216:
3195:
3190:
3189:
3164:
3159:
3158:
3131:
3120:
3119:
3068:
3046:
3041:
3040:
3017:
3016:
2974:
2973:
2972:. A collection
2943:
2905:
2887:
2886:
2863:
2862:
2832:
2811:
2810:
2788:
2787:
2749:
2748:
2718:
2667:
2664:
2663:
2625:
2624:
2575:
2567:
2566:
2559:Komiya covering
2522:
2521:
2507:
2456:
2455:
2434:
2429:
2428:
2389:
2388:
2367:
2366:
2347:
2346:
2315:
2314:
2293:
2272:
2271:
2239:
2238:
2201:
2200:
2179:
2174:
2173:
2153:, a collection
2119:
2094:
2069:
2064:
2063:
2024:
2023:
2004:
2003:
1972:
1971:
1946:
1945:
1918:
1913:
1912:
1893:
1892:
1863:
1862:
1843:
1842:
1812:
1791:
1790:
1766:
1765:
1738:
1733:
1732:
1713:
1712:
1677:
1672:
1671:
1635:
1588:
1571:
1570:
1548:
1547:
1524:
1523:
1488:
1487:
1460:
1455:
1454:
1429:
1428:
1398:
1393:
1392:
1387:closed sets, a
1362:
1358:
1327:
1326:
1323:Sperner's lemma
1321:, generalizing
1289:
1284:
1283:
1259:
1254:
1253:
1232:
1213:
1208:
1207:
1177:
1082:
1081:
1059:
1058:
955:
954:
944:
940:
937:
935:Generalizations
905:Sperner's lemma
875:
820:
807:
794:
789:
788:
760:
755:
754:
733:
728:
727:
706:
701:
700:
679:
674:
673:
652:
647:
646:
625:
620:
619:
595:
590:
589:
568:
563:
562:
541:
536:
535:
511:
498:
485:
480:
479:
458:
453:
452:
427:
426:
423:
388:
362:
361:
326:
305:
304:
281:
280:
233:
232:
207:
188:
183:
182:
144:
143:
101:
100:
73:
68:
67:
64:
52:Sperner's lemma
23:
22:
15:
12:
11:
5:
4002:
4000:
3992:
3991:
3986:
3981:
3971:
3970:
3967:
3966:
3957:
3956:External links
3954:
3951:
3950:
3913:(3): 627–637.
3891:
3846:
3801:
3766:(3): 473–477.
3750:
3723:
3704:(2): 361–367.
3688:
3669:(3): 463–466.
3650:
3615:
3602:(2): 297–299.
3577:
3542:
3535:
3515:
3477:
3439:
3413:(4): 346–354,
3387:
3374:(1): 132–137,
3350:Kuratowski, C.
3336:
3335:
3333:
3330:
3329:
3328:
3319:
3316:
3307:
3304:
3291:
3288:
3284:
3280:
3277:
3274:
3271:
3268:
3264:
3260:
3257:
3235:
3232:
3229:
3225:
3202:
3198:
3171:
3167:
3144:
3141:
3138:
3134:
3130:
3127:
3092:
3089:
3086:
3083:
3080:
3075:
3071:
3067:
3064:
3061:
3058:
3053:
3049:
3000:
2997:
2994:
2984:
2981:
2961:
2958:
2955:
2950:
2946:
2942:
2939:
2936:
2933:
2923:
2920:
2917:
2912:
2908:
2904:
2901:
2870:
2859:
2858:
2847:
2844:
2839:
2835:
2829:
2826:
2823:
2819:
2806:is nonempty:
2795:
2775:
2772:
2769:
2759:
2756:
2730:
2725:
2721:
2715:
2712:
2709:
2699:
2696:
2690:
2687:
2684:
2681:
2677:
2673:
2670:
2651:
2648:
2645:
2635:
2632:
2612:
2609:
2606:
2603:
2593:
2590:
2587:
2582:
2578:
2574:
2542:
2539:
2536:
2506:
2503:
2472:
2468:
2464:
2441:
2437:
2425:
2424:
2412:
2409:
2405:
2401:
2397:
2386:
2374:
2354:
2334:
2331:
2328:
2325:
2322:
2300:
2296:
2292:
2288:
2284:
2280:
2255:
2251:
2247:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2186:
2182:
2147:
2146:
2134:
2131:
2126:
2122:
2118:
2115:
2112:
2107:
2104:
2101:
2097:
2093:
2090:
2087:
2082:
2079:
2076:
2072:
2060:
2057:
2054:
2037:
2034:
2031:
2011:
1991:
1988:
1985:
1982:
1979:
1959:
1956:
1953:
1933:
1930:
1925:
1921:
1900:
1876:
1873:
1870:
1861:of subsets of
1850:
1839:
1838:
1827:
1824:
1819:
1815:
1809:
1806:
1803:
1799:
1773:
1751:
1748:
1745:
1741:
1720:
1692:
1689:
1684:
1680:
1642:
1638:
1632:
1629:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1595:
1591:
1587:
1584:
1581:
1578:
1555:
1531:
1507:
1504:
1501:
1498:
1495:
1473:
1470:
1467:
1463:
1442:
1439:
1436:
1413:
1410:
1405:
1401:
1361:
1356:
1340:
1337:
1334:
1319:Ravindra Bapat
1296:
1292:
1266:
1262:
1239:
1235:
1231:
1228:
1225:
1220:
1216:
1206:is a covering
1176:
1173:
1150:
1149:
1137:
1134:
1129:
1126:
1123:
1120:
1115:
1111:
1105:
1100:
1097:
1094:
1090:
1066:
1044:
1039:
1035:
1031:
1028:
1025:
1020:
1015:
1011:
1007:
1004:
1001:
996:
991:
987:
983:
980:
977:
972:
967:
963:
943:
938:
936:
933:
930:
929:
924:
922:Tucker's lemma
919:
913:
912:
907:
902:
896:
895:
892:
889:
874:
871:
863:
862:
859:
856:
827:
823:
819:
814:
810:
806:
801:
797:
785:
784:
781:
767:
763:
740:
736:
713:
709:
686:
682:
659:
655:
632:
628:
616:
602:
598:
575:
571:
548:
544:
518:
514:
510:
505:
501:
497:
492:
488:
465:
461:
440:
437:
434:
422:
419:
418:
417:
406:
403:
400:
395:
391:
385:
380:
377:
374:
370:
333:
329:
323:
320:
317:
313:
288:
264:
261:
258:
255:
252:
249:
246:
243:
240:
214:
210:
206:
203:
200:
195:
191:
163:
160:
157:
154:
151:
120:
117:
114:
111:
108:
86:
83:
80:
76:
63:
60:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4001:
3990:
3987:
3985:
3982:
3980:
3977:
3976:
3974:
3964:
3960:
3959:
3955:
3946:
3942:
3938:
3934:
3930:
3926:
3921:
3916:
3912:
3908:
3907:
3906:Combinatorica
3902:
3895:
3892:
3887:
3883:
3879:
3875:
3870:
3865:
3861:
3857:
3850:
3847:
3842:
3838:
3834:
3830:
3825:
3820:
3816:
3812:
3805:
3802:
3797:
3793:
3789:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3754:
3751:
3746:
3742:
3738:
3734:
3727:
3724:
3719:
3715:
3711:
3707:
3703:
3699:
3692:
3689:
3684:
3680:
3676:
3672:
3668:
3664:
3657:
3655:
3651:
3646:
3642:
3638:
3634:
3630:
3626:
3619:
3616:
3610:
3605:
3601:
3597:
3593:
3586:
3584:
3582:
3578:
3573:
3569:
3565:
3561:
3557:
3553:
3546:
3543:
3538:
3536:9789814467896
3532:
3528:
3527:
3519:
3516:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3481:
3478:
3473:
3469:
3465:
3461:
3457:
3453:
3446:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3407:
3402:
3398:
3391:
3388:
3382:
3377:
3373:
3370:(in German),
3369:
3368:
3363:
3361:
3355:
3351:
3347:
3341:
3338:
3331:
3326:
3322:
3321:
3317:
3315:
3313:
3305:
3303:
3286:
3282:
3278:
3275:
3272:
3269:
3266:
3262:
3258:
3233:
3230:
3227:
3200:
3196:
3187:
3169:
3165:
3142:
3139:
3136:
3128:
3125:
3116:
3114:
3110:
3106:
3087:
3084:
3081:
3078:
3073:
3069:
3062:
3059:
3056:
3051:
3047:
3014:
2995:
2982:
2979:
2956:
2953:
2948:
2944:
2940:
2934:
2921:
2918:
2915:
2910:
2906:
2899:
2884:
2868:
2842:
2837:
2833:
2827:
2824:
2821:
2817:
2809:
2808:
2807:
2793:
2770:
2757:
2754:
2747:
2744:, there is a
2743:
2728:
2723:
2719:
2710:
2697:
2694:
2688:
2685:
2682:
2679:
2675:
2671:
2668:
2646:
2633:
2630:
2604:
2591:
2588:
2585:
2580:
2576:
2564:
2560:
2556:
2537:
2519:
2515:
2511:
2504:
2502:
2500:
2495:
2493:
2488:
2485:
2470:
2466:
2462:
2439:
2435:
2407:
2403:
2399:
2395:
2387:
2372:
2352:
2329:
2323:
2320:
2298:
2294:
2290:
2286:
2282:
2278:
2270:
2269:
2268:
2253:
2249:
2245:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2184:
2180:
2170:
2168:
2165:and edge-set
2164:
2160:
2156:
2152:
2132:
2129:
2124:
2120:
2116:
2113:
2110:
2105:
2102:
2099:
2095:
2091:
2088:
2085:
2080:
2077:
2074:
2070:
2061:
2058:
2055:
2051:
2050:
2049:
2035:
2032:
2029:
2009:
1986:
1980:
1977:
1957:
1954:
1951:
1931:
1928:
1923:
1919:
1898:
1890:
1871:
1848:
1822:
1817:
1813:
1807:
1804:
1801:
1797:
1789:
1788:
1787:
1785:
1771:
1746:
1739:
1718:
1711:
1704:
1690:
1687:
1682:
1678:
1669:
1664:
1662:
1656:
1640:
1636:
1630:
1627:
1624:
1620:
1616:
1607:
1604:
1601:
1598:
1593:
1589:
1579:
1576:
1567:
1553:
1545:
1529:
1521:
1502:
1496:
1493:
1471:
1468:
1465:
1437:
1426:
1411:
1408:
1403:
1399:
1390:
1389:KKMS covering
1386:
1381:
1379:
1375:
1371:
1370:Lloyd Shapley
1367:
1357:
1355:
1352:
1338:
1335:
1332:
1324:
1320:
1317:A theorem of
1315:
1312:
1294:
1290:
1280:
1264:
1260:
1237:
1233:
1229:
1226:
1223:
1218:
1214:
1205:
1200:
1198:
1194:
1193:connected set
1190:
1181:
1174:
1172:
1170:
1165:
1163:
1157:
1153:
1132:
1124:
1118:
1113:
1109:
1103:
1098:
1095:
1092:
1088:
1080:
1079:
1078:
1064:
1042:
1037:
1033:
1029:
1026:
1023:
1018:
1013:
1009:
1005:
1002:
999:
994:
989:
985:
981:
978:
975:
970:
965:
961:
952:
948:
939:
934:
928:
925:
923:
920:
918:
915:
914:
911:
908:
906:
903:
901:
898:
897:
894:Set covering
893:
891:Combinatorics
890:
887:
886:
883:
880:
872:
870:
866:
860:
857:
854:
853:
852:
845:
841:
825:
821:
817:
812:
808:
804:
799:
795:
782:
765:
761:
738:
734:
711:
707:
684:
680:
657:
653:
630:
626:
617:
600:
596:
573:
569:
546:
542:
534:
533:
532:
516:
512:
508:
503:
499:
495:
490:
486:
463:
438:
435:
432:
420:
404:
398:
393:
389:
383:
378:
375:
372:
368:
360:
359:
358:
356:
354:
347:
331:
327:
321:
318:
315:
311:
302:
286:
278:
259:
256:
253:
250:
247:
241:
238:
230:
212:
208:
204:
201:
198:
193:
189:
180:
175:
161:
158:
155:
152:
149:
141:
138:
134:
131:-dimensional
115:
112:
109:
84:
81:
78:
61:
59:
57:
53:
48:
46:
42:
38:
34:
30:
19:
18:K-k-m theorem
3910:
3904:
3894:
3859:
3855:
3849:
3814:
3810:
3804:
3763:
3759:
3753:
3736:
3732:
3726:
3701:
3697:
3691:
3666:
3662:
3628:
3624:
3618:
3599:
3595:
3555:
3551:
3545:
3525:
3518:
3493:
3489:
3486:Bapat, R. B.
3480:
3455:
3451:
3410:
3404:
3390:
3371:
3365:
3359:
3340:
3309:
3117:
3112:
3104:
3012:
2882:
2860:
2745:
2741:
2562:
2558:
2554:
2517:
2508:
2496:
2491:
2489:
2486:
2426:
2267:as follows:
2171:
2166:
2162:
2158:
2154:
2148:
1888:
1840:
1709:
1707:
1705:
1667:
1665:
1660:
1658:
1569:
1388:
1384:
1382:
1366:KKMS theorem
1365:
1363:
1359:KKMS theorem
1353:
1316:
1310:
1281:
1252:in which no
1203:
1201:
1196:
1188:
1186:
1168:
1166:
1159:
1155:
1151:
950:
945:
909:
876:
867:
864:
850:
786:
424:
352:
350:
348:
179:KKM covering
178:
176:
136:
65:
49:
45:Mazurkiewicz
28:
26:
3963:Planet Math
3558:: 108–116.
3346:Knaster, B.
1784:is nonempty
1566:, that is:
1520:convex hull
1486:). For any
1425:closed sets
531:such that:
277:convex hull
229:closed sets
142:labeled as
3973:Categories
3920:1710.07722
3869:1505.07629
3824:1512.04612
3739:(2): 125.
3631:(3): 467.
3332:References
3186:barycenter
3011:is called
2423:otherwise.
1887:is called
1542:should be
947:David Gale
41:Kuratowski
3945:119176249
3937:1439-6912
3886:119695004
3841:119619991
3796:120862302
3780:0938-2259
3718:122754557
3683:123150937
3572:121027709
3472:154888988
3458:: 61–64.
3273:…
3231:−
3224:Δ
3140:−
3133:Δ
3085:∈
3063:
3057:∈
2983:⊆
2954:∈
2922:∈
2846:∅
2843:≠
2825:∈
2818:⋂
2758:⊆
2698:∈
2683:⊆
2676:⋃
2672:⊆
2634:∈
2592:∈
2514:polytopes
2499:open sets
2492:partnered
2411:∅
2313:whenever
2219:…
1981:∈
1955:∈
1944:to every
1929:≥
1826:∅
1823:≠
1805:∈
1798:⋂
1688:−
1628:⊆
1621:⋃
1617:⊆
1605:∈
1580:
1497:⊆
1469:−
1462:Δ
1409:−
1391:contains
1374:economics
1227:…
1189:connector
1136:∅
1133:≠
1119:π
1089:⋂
1065:π
1027:…
1003:…
979:…
460:Δ
402:∅
399:≠
369:⋂
319:∈
312:⋃
254:…
242:⊆
202:…
156:…
113:−
82:−
75:Δ
62:Statement
3788:25054778
3645:15004516
3399:(2013),
3356:(1929),
3318:See also
3312:homotopy
3013:balanced
2471:′
2404:′
2287:′
2254:′
2048:. Then:
1889:balanced
882:column.
140:vertices
3510:5325605
3435:3035127
3184:is the
1544:covered
421:Example
357:, i.e:
301:covered
133:simplex
37:Knaster
3989:Lemmas
3943:
3935:
3884:
3839:
3794:
3786:
3778:
3716:
3681:
3643:
3570:
3533:
3508:
3470:
3433:
3425:
2692:
2516:. Let
2199:, for
1518:, the
275:, the
99:be an
3941:S2CID
3915:arXiv
3882:S2CID
3864:arXiv
3837:S2CID
3819:arXiv
3792:S2CID
3784:JSTOR
3714:S2CID
3679:S2CID
3641:S2CID
3568:S2CID
3506:S2CID
3468:S2CID
3423:JSTOR
3107:is a
2988:Faces
2927:Faces
2763:Faces
2703:Faces
2639:Faces
2597:Faces
2557:. A
2530:Faces
425:When
135:with
3933:ISSN
3776:ISSN
3531:ISBN
3157:and
3060:conv
3039:iff
1577:conv
1364:The
753:and
699:and
645:and
66:Let
43:and
27:The
3925:doi
3874:doi
3829:doi
3768:doi
3741:doi
3706:doi
3671:doi
3633:doi
3604:doi
3560:doi
3498:doi
3460:doi
3415:doi
3411:120
3376:doi
3302:).
2561:of
2149:In
1731:of
1668:not
1351:).
303:by
299:is
227:of
3975::
3939:.
3931:.
3923:.
3911:39
3909:.
3903:.
3880:.
3872:.
3860:16
3858:.
3835:.
3827:.
3815:19
3813:.
3790:.
3782:.
3774:.
3762:.
3737:29
3735:.
3712:.
3702:10
3700:.
3677:.
3665:.
3653:^
3639:.
3627:.
3600:81
3598:.
3594:.
3580:^
3566:.
3554:.
3504:.
3494:44
3492:.
3466:.
3456:13
3454:.
3442:^
3431:MR
3429:,
3421:,
3409:,
3403:,
3372:14
3364:,
3352:;
3348:;
3314:.
3115:.
2662::
2494:.
2385:).
2053:}.
1786::
1655:.
1380:.
1202:A
1187:A
1164:.
346:.
177:A
174:.
58:.
47:.
39:,
3965:.
3947:.
3927::
3917::
3888:.
3876::
3866::
3843:.
3831::
3821::
3798:.
3770::
3764:4
3747:.
3743::
3720:.
3708::
3685:.
3673::
3667:4
3647:.
3635::
3629:4
3612:.
3606::
3574:.
3562::
3556:1
3539:.
3512:.
3500::
3474:.
3462::
3417::
3385:.
3378::
3360:n
3327:.
3290:)
3287:n
3283:/
3279:1
3276:,
3270:,
3267:n
3263:/
3259:1
3256:(
3234:1
3228:n
3201:P
3197:b
3170:F
3166:b
3143:1
3137:n
3129:=
3126:P
3113:B
3105:P
3091:}
3088:B
3082:F
3079::
3074:F
3070:b
3066:{
3052:P
3048:b
3025:b
2999:)
2996:P
2993:(
2980:B
2960:}
2957:F
2949:F
2945:b
2941:,
2938:)
2935:P
2932:(
2919:F
2916::
2911:F
2907:b
2903:{
2900:=
2895:b
2883:P
2869:B
2838:F
2834:C
2828:B
2822:F
2794:B
2774:)
2771:P
2768:(
2755:B
2742:P
2729:.
2724:G
2720:C
2714:)
2711:P
2708:(
2695:G
2689:,
2686:F
2680:G
2669:F
2650:)
2647:P
2644:(
2631:F
2611:}
2608:)
2605:P
2602:(
2589:F
2586::
2581:F
2577:C
2573:{
2563:P
2555:P
2541:)
2538:P
2535:(
2518:P
2467:J
2463:C
2440:i
2436:C
2408:=
2400:J
2396:C
2373:i
2353:J
2345:(
2333:}
2330:i
2327:{
2324:=
2321:J
2299:i
2295:C
2291:=
2283:J
2279:C
2250:J
2246:C
2225:n
2222:,
2216:,
2213:1
2210:=
2207:i
2185:i
2181:C
2167:B
2163:V
2159:V
2155:B
2145:.
2133:1
2130:=
2125:1
2121:w
2117:,
2114:1
2111:=
2106:3
2103:,
2100:2
2096:w
2092:,
2089:0
2086:=
2081:2
2078:,
2075:1
2071:w
2036:3
2033:=
2030:n
2010:i
1990:]
1987:n
1984:[
1978:i
1958:B
1952:J
1932:0
1924:J
1920:w
1899:B
1875:]
1872:n
1869:[
1849:B
1818:J
1814:C
1808:B
1802:J
1772:B
1750:]
1747:n
1744:[
1740:2
1719:B
1691:1
1683:n
1679:2
1661:n
1641:J
1637:C
1631:I
1625:J
1614:)
1611:}
1608:I
1602:i
1599::
1594:i
1590:v
1586:{
1583:(
1554:I
1530:I
1506:]
1503:n
1500:[
1494:I
1472:1
1466:n
1441:]
1438:n
1435:[
1412:1
1404:n
1400:2
1385:n
1339:3
1336:=
1333:n
1311:n
1295:i
1291:C
1265:i
1261:C
1238:n
1234:C
1230:,
1224:,
1219:1
1215:C
1197:n
1169:n
1148:.
1128:)
1125:i
1122:(
1114:i
1110:C
1104:n
1099:1
1096:=
1093:i
1043:n
1038:n
1034:C
1030:,
1024:,
1019:n
1014:1
1010:C
1006:,
1000:,
995:1
990:n
986:C
982:,
976:,
971:1
966:1
962:C
951:n
826:3
822:C
818:,
813:2
809:C
805:,
800:1
796:C
780:.
766:1
762:C
739:3
735:C
712:3
708:C
685:2
681:C
658:2
654:C
631:1
627:C
601:3
597:C
574:2
570:C
547:1
543:C
517:3
513:C
509:,
504:2
500:C
496:,
491:1
487:C
464:2
439:3
436:=
433:n
405:.
394:i
390:C
384:n
379:1
376:=
373:i
353:n
332:i
328:C
322:I
316:i
287:I
263:}
260:n
257:,
251:,
248:1
245:{
239:I
213:n
209:C
205:,
199:,
194:1
190:C
162:n
159:,
153:,
150:1
137:n
119:)
116:1
110:n
107:(
85:1
79:n
20:)
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