355:
on the number of incidences that are possible between given numbers of points and lines in the plane, follows by constructing a graph whose vertices are the points and whose edges are the segments of lines between incident points. If there were more incidences than the Szemerédi–Trotter bound, this
323:
refers to the minimum number of pairs of edges that each determine one crossing, whereas the crossing number simply refers to the minimum number of crossings. (This distinction is necessary since some authors assume that in a proper drawing no two edges cross more than once.)
867:
crossings. We may assume that any two edges in this diagram with a common vertex are disjoint, otherwise we could interchange the intersecting parts of the two edges and reduce the crossing number by one. Thus every crossing in this diagram involves four distinct vertices of
1590:
1200:
1059:
219:
779:
961:
677:
1445:
321:
443:
1366:
1303:
1258:
1226:
1465:
1473:
1814:
1845:
360:, that if a finite point set does not have a linear number of collinear points, then it determines a quadratic number of distinct lines. Similarly,
1121:
356:
graph would necessarily have more crossings than the total number of pairs of lines, an impossibility. The inequality can also be used to prove
997:
157:
1943:
1897:
1775:
2006:
692:
348:
1669:
893:
1773:; Radoičić, Radoš; Tardos, Gábor; Tóth, Géza (2006), "Improving the crossing lemma by finding more crossings in sparse graphs",
623:
1371:
138:
36:
278:
60:
357:
40:
398:
1308:
149:
2016:
81:
1263:
2011:
48:
1231:
1612:
1091:
527:
344:
89:
1645:
101:
44:
1981:
1870:
1752:
1704:
1678:
228:
is the best known to date, and is due to
Ackerman. For earlier results with weaker constants see
1616:
97:
1211:
1952:
1906:
1854:
1823:
1784:
1736:
1688:
1623:, North-Holland Mathematics Studies, vol. 60, North-Holland, Amsterdam, pp. 9–12,
1205:
365:
1966:
1920:
1866:
1798:
1748:
1700:
1628:
1962:
1916:
1862:
1794:
1744:
1696:
1624:
1608:
85:
1843:
Székely, L. A. (1997), "Crossing numbers and hard Erdős problems in discrete geometry",
1450:
683:
343:
realized that this inequality yielded very simple proofs of some important theorems in
1980:
Adiprasito, Karim (2018-12-26), "Combinatorial
Lefschetz theorems beyond positivity",
2000:
1934:
1770:
1727:
1722:
1667:
Ackerman, Eyal (2019), "On topological graphs with at most four crossings per edge",
93:
20:
1874:
1756:
1708:
1938:
543:
484:
480:
113:
1585:{\displaystyle {\frac {f_{d}^{d+2}(\Delta )}{(d+3)^{d+2}f_{d-1}^{d+1}(\Delta )}}.}
1692:
855:, since every crossing involves four vertices. To see this consider a diagram of
539:
352:
332:
The motivation of
Leighton in studying crossing numbers was for applications to
32:
1858:
1789:
1888:
460:
crossings. Each of these crossings can be removed by removing an edge from
361:
1828:
1195:{\displaystyle \operatorname {cr} (G)\geq c_{r}{\frac {e^{r+2}}{n^{r+1}}}.}
1957:
1911:
1812:
Pach, János; Tóth, Géza (2000), "Which crossing number is it anyway?",
1740:
1986:
1054:{\displaystyle \operatorname {cr} (G)\geq {\frac {e^{3}}{64n^{2}}}.}
214:{\displaystyle \operatorname {cr} (G)\geq {\frac {e^{3}}{29n^{2}}}.}
1683:
1725:; Tóth, Géza (1997), "Graphs drawn with few crossings per edge",
333:
77:
526:
To obtain the actual crossing number inequality, we now use a
112:
The crossing number inequality states that, for an undirected
1228:
is a simplicial complex that is mapped piecewise-linearly to
774:{\displaystyle \mathbf {E} \geq \mathbf {E} -3\mathbf {E} .}
1652:, Foundations of Computing Series, Cambridge, MA: MIT Press
1064:
A slight refinement of this argument allows one to replace
956:{\displaystyle p^{4}\operatorname {cr} (G)\geq p^{2}e-3pn.}
255:
It is important to distinguish between the crossing number
51:
of the graph. It states that, for graphs where the number
672:{\displaystyle \operatorname {cr} _{H}\geq e_{H}-3n_{H}.}
617:. By the preliminary crossing number inequality, we have
1440:{\displaystyle f_{d}(\Delta )>(d+3)f_{d-1}(\Delta )}
597:
denote the number of edges, vertices and crossings of
573:
if and only if its two vertices were chosen to lie in
316:{\displaystyle {\text{pair-cr}}(G)\leq {\text{cr}}(G)}
233:
1941:; Tóth, G. (2000), "New bounds on crossing numbers",
1476:
1453:
1374:
1311:
1266:
1234:
1214:
1124:
1000:
896:
695:
626:
401:
281:
160:
380:
We first give a preliminary estimate: for any graph
1584:
1459:
1439:
1360:
1297:
1252:
1220:
1194:
1115:demonstrated an improvement of this inequality to
1053:
955:
773:
671:
437:
315:
213:
1208:showed a generalization to higher dimensions: If
438:{\displaystyle \operatorname {cr} (G)\geq e-3n.}
55:of edges is sufficiently larger than the number
1112:
542:parameter to be chosen later, and construct a
59:of vertices, the crossing number is at least
8:
841:. Finally, every crossing in the diagram of
16:Drawings of dense graphs have many crossings
1447:, then the number of pairwise intersecting
1640:
1638:
501:, and the claim follows. (In fact we have
479:vertices with no crossings, and is thus a
1985:
1956:
1910:
1827:
1815:Journal of Combinatorial Theory, Series B
1788:
1682:
1555:
1544:
1528:
1489:
1484:
1477:
1475:
1452:
1416:
1379:
1373:
1361:{\displaystyle f_{d-1}(\Delta )\ \ (d-1)}
1316:
1310:
1271:
1265:
1241:
1236:
1233:
1213:
1175:
1159:
1153:
1147:
1123:
1039:
1025:
1019:
999:
929:
901:
895:
759:
747:
732:
720:
708:
696:
694:
660:
644:
631:
625:
464:. Thus we can find a graph with at least
400:
299:
282:
280:
199:
185:
179:
159:
1846:Combinatorics, Probability and Computing
336:design in theoretical computer science.
272:
229:
1600:
340:
84:, and was discovered independently by
824:since both endpoints need to stay in
448:To prove this, consider a diagram of
7:
1891:(1998), "Improved bounds for planar
1662:
1660:
1621:Theory and practice of combinatorics
1944:Discrete and Computational Geometry
1898:Discrete and Computational Geometry
1776:Discrete and Computational Geometry
1619:(1982), "Crossing-free subgraphs",
1298:{\displaystyle f_{d}(\Delta )\ \ d}
1570:
1504:
1431:
1388:
1331:
1280:
1215:
810:. Similarly, each of the edges in
244:, but at the expense of replacing
14:
1253:{\displaystyle \mathbf {R} ^{2d}}
364:used it to prove upper bounds on
263:and the pairwise crossing number
43:, as a function of the number of
1237:
991:), we obtain after some algebra
748:
721:
697:
37:minimum number of edge crossings
1467:-dimensional faces is at least
1113:Pach, Spencer & Tóth (2000)
561:independently with probability
275:, the pairwise crossing number
1573:
1567:
1525:
1512:
1507:
1501:
1434:
1428:
1409:
1397:
1391:
1385:
1355:
1343:
1334:
1328:
1283:
1277:
1137:
1131:
1013:
1007:
919:
913:
765:
752:
738:
725:
714:
701:
414:
408:
310:
304:
293:
287:
173:
167:
39:in a plane drawing of a given
1:
1895:-sets and related problems",
1368:-dimensional faces such that
1693:10.1016/j.comgeo.2019.101574
553:by allowing each vertex of
248:with the worse constant of
2033:
565:, and allowing an edge of
25:crossing number inequality
1859:10.1017/S0963548397002976
1790:10.1007/s00454-006-1264-9
1650:Complexity Issues in VLSI
349:Szemerédi–Trotter theorem
2007:Topological graph theory
1305:-dimensional faces and
1221:{\displaystyle \Delta }
981:(since we assumed that
76:It has applications in
1829:10.1006/jctb.2000.1978
1670:Computational Geometry
1586:
1461:
1441:
1362:
1299:
1254:
1222:
1196:
1055:
957:
775:
673:
613:contains a diagram of
601:, respectively. Since
528:probabilistic argument
439:
317:
273:Pach & Tóth (2000)
230:Pach & Tóth (1997)
215:
82:combinatorial geometry
19:In the mathematics of
1958:10.1007/s004540010011
1587:
1462:
1442:
1363:
1300:
1255:
1223:
1197:
1056:
958:
776:
674:
440:
318:
216:
108:Statement and history
1474:
1451:
1372:
1309:
1264:
1232:
1212:
1122:
998:
894:
693:
624:
399:
347:. For instance, the
279:
158:
1566:
1500:
1912:10.1007/PL00009354
1741:10.1007/BF01215922
1615:; Newborn, M. M.;
1582:
1540:
1480:
1457:
1437:
1358:
1295:
1250:
1218:
1192:
1051:
953:
845:has a probability
814:has a probability
792:had a probability
784:Since each of the
771:
669:
487:we must then have
452:which has exactly
435:
345:incidence geometry
313:
240:can be lowered to
234:Pach et al. (2006)
211:
1577:
1460:{\displaystyle d}
1342:
1339:
1291:
1288:
1187:
1046:
609:, the diagram of
605:is a subgraph of
302:
285:
206:
2024:
1992:
1990:
1989:
1977:
1971:
1969:
1960:
1931:
1925:
1923:
1914:
1894:
1885:
1879:
1877:
1840:
1834:
1832:
1831:
1809:
1803:
1801:
1792:
1767:
1761:
1759:
1719:
1713:
1711:
1686:
1664:
1655:
1653:
1642:
1633:
1631:
1605:
1591:
1589:
1588:
1583:
1578:
1576:
1565:
1554:
1539:
1538:
1510:
1499:
1488:
1478:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1427:
1426:
1384:
1383:
1367:
1365:
1364:
1359:
1340:
1337:
1327:
1326:
1304:
1302:
1301:
1296:
1289:
1286:
1276:
1275:
1259:
1257:
1256:
1251:
1249:
1248:
1240:
1227:
1225:
1224:
1219:
1201:
1199:
1198:
1193:
1188:
1186:
1185:
1170:
1169:
1154:
1152:
1151:
1110:
1100:
1090:For graphs with
1081:
1071:
1067:
1060:
1058:
1057:
1052:
1047:
1045:
1044:
1043:
1030:
1029:
1020:
990:
980:
962:
960:
959:
954:
934:
933:
906:
905:
886:
871:
866:
858:
854:
851:of remaining in
850:
844:
840:
827:
823:
820:of remaining in
819:
813:
809:
799:
795:
791:
787:
780:
778:
777:
772:
764:
763:
751:
737:
736:
724:
713:
712:
700:
678:
676:
675:
670:
665:
664:
649:
648:
636:
635:
616:
612:
608:
604:
600:
596:
587:
576:
572:
568:
564:
560:
556:
552:
548:
537:
522:
515:
500:
478:
474:
463:
459:
451:
444:
442:
441:
436:
391:
387:
383:
322:
320:
319:
314:
303:
300:
286:
283:
270:
262:
251:
247:
243:
239:
227:
220:
218:
217:
212:
207:
205:
204:
203:
190:
189:
180:
147:
136:
127:edges such that
126:
122:
118:
72:
58:
54:
2032:
2031:
2027:
2026:
2025:
2023:
2022:
2021:
1997:
1996:
1995:
1979:
1978:
1974:
1933:
1932:
1928:
1892:
1887:
1886:
1882:
1842:
1841:
1837:
1811:
1810:
1806:
1769:
1768:
1764:
1721:
1720:
1716:
1666:
1665:
1658:
1644:
1643:
1636:
1607:
1606:
1602:
1598:
1524:
1511:
1479:
1472:
1471:
1449:
1448:
1412:
1375:
1370:
1369:
1312:
1307:
1306:
1267:
1262:
1261:
1235:
1230:
1229:
1210:
1209:
1171:
1155:
1143:
1120:
1119:
1102:
1095:
1088:
1073:
1069:
1065:
1035:
1031:
1021:
996:
995:
982:
967:
925:
897:
892:
891:
873:
869:
860:
856:
852:
846:
842:
829:
825:
821:
815:
811:
801:
797:
793:
789:
785:
755:
728:
704:
691:
690:
656:
640:
627:
622:
621:
614:
610:
606:
602:
598:
595:
589:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
544:random subgraph
531:
517:
502:
488:
485:Euler's formula
476:
465:
461:
453:
449:
397:
396:
392:edges, we have
389:
385:
381:
378:
330:
277:
276:
264:
256:
249:
245:
241:
237:
236:. The constant
225:
195:
191:
181:
156:
155:
141:
139:crossing number
128:
124:
120:
116:
110:
64:
56:
52:
17:
12:
11:
5:
2030:
2028:
2020:
2019:
2014:
2009:
1999:
1998:
1994:
1993:
1972:
1951:(4): 623–644,
1926:
1905:(3): 373–382,
1880:
1853:(3): 353–358,
1835:
1822:(2): 225–246,
1804:
1783:(4): 527–552,
1762:
1735:(3): 427–439,
1714:
1677:: 101574, 31,
1656:
1634:
1599:
1597:
1594:
1593:
1592:
1581:
1575:
1572:
1569:
1564:
1561:
1558:
1553:
1550:
1547:
1543:
1537:
1534:
1531:
1527:
1523:
1520:
1517:
1514:
1509:
1506:
1503:
1498:
1495:
1492:
1487:
1483:
1456:
1436:
1433:
1430:
1425:
1422:
1419:
1415:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1382:
1378:
1357:
1354:
1351:
1348:
1345:
1336:
1333:
1330:
1325:
1322:
1319:
1315:
1294:
1285:
1282:
1279:
1274:
1270:
1247:
1244:
1239:
1217:
1203:
1202:
1191:
1184:
1181:
1178:
1174:
1168:
1165:
1162:
1158:
1150:
1146:
1142:
1139:
1136:
1133:
1130:
1127:
1087:
1084:
1062:
1061:
1050:
1042:
1038:
1034:
1028:
1024:
1018:
1015:
1012:
1009:
1006:
1003:
966:Now if we set
964:
963:
952:
949:
946:
943:
940:
937:
932:
928:
924:
921:
918:
915:
912:
909:
904:
900:
782:
781:
770:
767:
762:
758:
754:
750:
746:
743:
740:
735:
731:
727:
723:
719:
716:
711:
707:
703:
699:
680:
679:
668:
663:
659:
655:
652:
647:
643:
639:
634:
630:
591:
584:
580:
446:
445:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
377:
374:
358:Beck's theorem
341:Székely (1997)
329:
326:
312:
309:
306:
298:
295:
292:
289:
271:. As noted by
222:
221:
210:
202:
198:
194:
188:
184:
178:
175:
172:
169:
166:
163:
109:
106:
29:crossing lemma
15:
13:
10:
9:
6:
4:
3:
2:
2029:
2018:
2017:Graph drawing
2015:
2013:
2010:
2008:
2005:
2004:
2002:
1988:
1983:
1976:
1973:
1968:
1964:
1959:
1954:
1950:
1946:
1945:
1940:
1936:
1930:
1927:
1922:
1918:
1913:
1908:
1904:
1900:
1899:
1890:
1884:
1881:
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1847:
1839:
1836:
1830:
1825:
1821:
1817:
1816:
1808:
1805:
1800:
1796:
1791:
1786:
1782:
1778:
1777:
1772:
1766:
1763:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1729:
1728:Combinatorica
1724:
1718:
1715:
1710:
1706:
1702:
1698:
1694:
1690:
1685:
1680:
1676:
1672:
1671:
1663:
1661:
1657:
1651:
1647:
1641:
1639:
1635:
1630:
1626:
1622:
1618:
1617:Szemerédi, E.
1614:
1610:
1604:
1601:
1595:
1579:
1562:
1559:
1556:
1551:
1548:
1545:
1541:
1535:
1532:
1529:
1521:
1518:
1515:
1496:
1493:
1490:
1485:
1481:
1470:
1469:
1468:
1454:
1423:
1420:
1417:
1413:
1406:
1403:
1400:
1394:
1380:
1376:
1352:
1349:
1346:
1323:
1320:
1317:
1313:
1292:
1272:
1268:
1260:, and it has
1245:
1242:
1207:
1189:
1182:
1179:
1176:
1172:
1166:
1163:
1160:
1156:
1148:
1144:
1140:
1134:
1128:
1125:
1118:
1117:
1116:
1114:
1109:
1105:
1099:
1093:
1085:
1083:
1080:
1076:
1048:
1040:
1036:
1032:
1026:
1022:
1016:
1010:
1004:
1001:
994:
993:
992:
989:
985:
978:
974:
970:
950:
947:
944:
941:
938:
935:
930:
926:
922:
916:
910:
907:
902:
898:
890:
889:
888:
884:
880:
876:
872:. Therefore,
864:
849:
839:
836:
832:
818:
808:
804:
768:
760:
756:
744:
741:
733:
729:
717:
709:
705:
689:
688:
687:
685:
666:
661:
657:
653:
650:
645:
641:
637:
632:
628:
620:
619:
618:
594:
545:
541:
535:
529:
524:
520:
513:
509:
505:
499:
495:
491:
486:
482:
472:
468:
457:
432:
429:
426:
423:
420:
417:
411:
405:
402:
395:
394:
393:
388:vertices and
375:
373:
371:
369:
363:
359:
354:
350:
346:
342:
337:
335:
327:
325:
307:
296:
290:
274:
268:
260:
253:
235:
231:
224:The constant
208:
200:
196:
192:
186:
182:
176:
170:
164:
161:
154:
153:
152:
151:
145:
140:
135:
131:
123:vertices and
115:
107:
105:
103:
99:
95:
91:
87:
83:
79:
74:
71:
67:
62:
50:
46:
42:
38:
34:
30:
26:
22:
21:graph drawing
2012:Inequalities
1987:1812.10454v3
1975:
1948:
1942:
1929:
1902:
1896:
1883:
1850:
1844:
1838:
1819:
1813:
1807:
1780:
1774:
1765:
1732:
1726:
1717:
1674:
1668:
1649:
1646:Leighton, T.
1620:
1603:
1204:
1107:
1103:
1097:
1094:larger than
1089:
1078:
1074:
1063:
987:
983:
976:
972:
968:
965:
887:and we have
882:
878:
874:
862:
847:
837:
834:
830:
828:, therefore
816:
806:
802:
796:of being in
788:vertices in
783:
684:expectations
681:
592:
533:
525:
518:
511:
507:
503:
497:
493:
489:
481:planar graph
470:
466:
455:
447:
379:
367:
338:
331:
328:Applications
266:
258:
254:
223:
143:
133:
129:
114:simple graph
111:
75:
69:
65:
61:proportional
28:
24:
18:
1939:Spencer, J.
1771:Pach, János
1723:Pach, János
1613:Chvátal, V.
540:probability
483:. But from
353:upper bound
80:design and
33:lower bound
2001:Categories
1889:Dey, T. K.
1684:1509.01932
1596:References
1206:Adiprasito
1086:Variations
800:, we have
686:we obtain
569:to lie in
557:to lie in
475:edges and
366:geometric
150:inequality
148:obeys the
1609:Ajtai, M.
1571:Δ
1549:−
1505:Δ
1432:Δ
1421:−
1389:Δ
1350:−
1332:Δ
1321:−
1281:Δ
1216:Δ
1141:≥
1129:
1017:≥
1005:
939:−
923:≥
911:
742:−
718:≥
651:−
638:≥
530:. We let
424:−
418:≥
406:
362:Tamal Dey
297:≤
177:≥
165:
98:Szemerédi
1935:Pach, J.
1875:36602807
1757:20480170
1709:16847443
1648:(1983),
1077:> 7.5
265:pair-cr(
102:Leighton
49:vertices
31:gives a
1967:1799605
1921:1608878
1867:1464571
1799:2267545
1749:1606052
1701:4010251
1629:0806962
682:Taking
532:0 <
339:Later,
284:pair-cr
100:and by
94:Newborn
90:Chvátal
35:on the
1965:
1919:
1873:
1865:
1797:
1755:
1747:
1707:
1699:
1627:
1341:
1338:
1290:
1287:
986:> 4
979:< 1
577:. Let
536:< 1
137:, the
132:> 7
96:, and
23:, the
1982:arXiv
1871:S2CID
1753:S2CID
1705:S2CID
1679:arXiv
1092:girth
1070:33.75
859:with
538:be a
510:) ≤ 3
506:− cr(
496:) ≤ 3
492:− cr(
469:− cr(
384:with
376:Proof
370:-sets
351:, an
119:with
86:Ajtai
45:edges
41:graph
1395:>
1101:and
1072:for
588:and
516:for
334:VLSI
232:and
78:VLSI
47:and
1953:doi
1907:doi
1855:doi
1824:doi
1785:doi
1737:doi
1689:doi
1106:≥ 4
1068:by
971:= 4
881:cr(
861:cr(
583:, n
549:of
523:).
521:≥ 3
514:− 6
454:cr(
257:cr(
142:cr(
63:to
27:or
2003::
1963:MR
1961:,
1949:24
1947:,
1937:;
1917:MR
1915:,
1903:19
1901:,
1869:,
1863:MR
1861:,
1849:,
1820:80
1818:,
1795:MR
1793:,
1781:36
1779:,
1751:,
1745:MR
1743:,
1733:17
1731:,
1703:,
1697:MR
1695:,
1687:,
1675:85
1673:,
1659:^
1637:^
1625:MR
1611:;
1126:cr
1111:,
1082:.
1066:64
1033:64
1002:cr
908:cr
877:=
833:=
807:pn
805:=
706:cr
629:cr
590:cr
403:cr
372:.
301:cr
252:.
250:64
246:29
226:29
193:29
162:cr
104:.
92:,
88:,
73:.
1991:.
1984::
1970:.
1955::
1924:.
1909::
1893:k
1878:.
1857::
1851:6
1833:.
1826::
1802:.
1787::
1760:.
1739::
1712:.
1691::
1681::
1654:.
1632:.
1580:.
1574:)
1568:(
1563:1
1560:+
1557:d
1552:1
1546:d
1542:f
1536:2
1533:+
1530:d
1526:)
1522:3
1519:+
1516:d
1513:(
1508:)
1502:(
1497:2
1494:+
1491:d
1486:d
1482:f
1455:d
1435:)
1429:(
1424:1
1418:d
1414:f
1410:)
1407:3
1404:+
1401:d
1398:(
1392:)
1386:(
1381:d
1377:f
1356:)
1353:1
1347:d
1344:(
1335:)
1329:(
1324:1
1318:d
1314:f
1293:d
1284:)
1278:(
1273:d
1269:f
1246:d
1243:2
1238:R
1190:.
1183:1
1180:+
1177:r
1173:n
1167:2
1164:+
1161:r
1157:e
1149:r
1145:c
1138:)
1135:G
1132:(
1108:n
1104:e
1098:r
1096:2
1079:n
1075:e
1049:.
1041:2
1037:n
1027:3
1023:e
1014:)
1011:G
1008:(
988:n
984:e
977:e
975:/
973:n
969:p
951:.
948:n
945:p
942:3
936:e
931:2
927:p
920:)
917:G
914:(
903:4
899:p
885:)
883:G
879:p
875:E
870:G
865:)
863:G
857:G
853:H
848:p
843:G
838:e
835:p
831:E
826:H
822:H
817:p
812:G
803:E
798:H
794:p
790:G
786:n
769:.
766:]
761:H
757:n
753:[
749:E
745:3
739:]
734:H
730:e
726:[
722:E
715:]
710:H
702:[
698:E
667:.
662:H
658:n
654:3
646:H
642:e
633:H
615:H
611:G
607:G
603:H
599:H
593:H
585:H
581:H
579:e
575:H
571:H
567:G
563:p
559:H
555:G
551:G
547:H
534:p
519:n
512:n
508:G
504:e
498:n
494:G
490:e
477:n
473:)
471:G
467:e
462:G
458:)
456:G
450:G
433:.
430:n
427:3
421:e
415:)
412:G
409:(
390:e
386:n
382:G
368:k
311:)
308:G
305:(
294:)
291:G
288:(
269:)
267:G
261:)
259:G
242:4
238:7
209:.
201:2
197:n
187:3
183:e
174:)
171:G
168:(
146:)
144:G
134:n
130:e
125:e
121:n
117:G
70:n
68:/
66:e
57:n
53:e
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