43:
104:
of a sunflower set is arranged in an intuitive way. Suppose the shared elements of a sunflower set are clumped together at the centre of the diagram, and the nonshared elements are distributed in a circular pattern around the shared elements. Then when the Venn diagram is completed, the lobe-shaped
2724:
5063:
2101:
5185:
5128:
1855:
143:
give successively weaker conditions which would imply the existence of a large sunflower in a given collection, with the latter being one of the most famous open problems of extremal combinatorics.
4206:
1608:
3325:
4585:
2942:
46:
A mathematical sunflower can be pictured as a flower. The kernel of the sunflower is the brown part in the middle, and each set of the sunflower is the union of a petal and the kernel.
1261:
2802:
3556:
2496:
3983:
5327:
5441:
3246:
3172:
3032:
1648:
3380:
1463:
581:
The study of sunflowers generally focuses on when set systems contain sunflowers, in particular, when a set system is sufficiently large to necessarily contain a sunflower.
1966:
5751:
3424:
5468:
5357:
5236:
4991:
4964:
4933:
4636:
1373:
4503:
2385:
399:
3651:
5905:
5408:
5011:
4849:
4826:
4796:
4772:
4752:
4730:
4359:
262:
131:
81:
5488:
5383:
5276:
4689:
3470:
3198:
3058:
2984:
2414:
2019:
1787:
1011:
936:
901:
875:
715:
617:
4413:
4386:
4302:
4255:
4010:
3915:
3842:
2488:
2441:
2338:
2231:
2178:
2131:
1993:
1882:
1721:
4532:
4076:
3888:
3815:
3769:
3743:
3677:
3610:
3584:
3124:
2204:
1674:
646:
5296:
5256:
5209:
5083:
4906:
4660:
4453:
4433:
4322:
4275:
4228:
4096:
4050:
4030:
3862:
3789:
3717:
3697:
3490:
3098:
3078:
2862:
2842:
2822:
2461:
2311:
2291:
2271:
2251:
2151:
1922:
1902:
1761:
1741:
1694:
1543:
1523:
1503:
1483:
1413:
1393:
1328:
1304:
1284:
1174:
1154:
1134:
1114:
1094:
1074:
1051:
1031:
976:
956:
835:
815:
795:
775:
755:
735:
689:
669:
563:
543:
519:
499:
479:
459:
439:
419:
367:
347:
327:
304:
284:
237:
217:
193:
169:
5665:
23:
For any sunflower size, does every set of uniformly sized sets which is of cardinality greater than some exponential in the set size contain a sunflower?
26:
5982:
5761:
5718:
5540:
5016:
4078:
partitions. This implies that at least two sets and not all the sets will have an element in common. Hence this is not a sunflower of
2027:
5805:
4870:
4052:
sets are selected from across multiple sets of the partition, then two must be selected from one partition since there are only
105:
subsets, which encircle the common elements and one or more unique elements, take on the appearance of the petals of a flower.
5133:
5088:
573:
subsets is also a sunflower. Similarly, a collection of sets each containing the same elements is also trivially a sunflower.
6159:
1795:
5924:
4878:
4596:
4107:
5850:
1548:
4032:
sets if in one of the disjoint partitions is sunflower free by assumption of H being sunflower free. Otherwise, if
3252:
36:
5513:
Alweiss, Ryan; Lovett, Shachar; Wu, Kewen; Zhang, Jiapeng (June 2020), "Improved bounds for the sunflower lemma",
4537:
4700:
90:
4692:
2867:
3200:. A 2021 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving that
2719:{\displaystyle |W|\leq \sum _{a\in A}|W_{a}|\leq |A|(f(k-1,r)-1)\leq k(r-1)f(k-1,r)-|A|\leq k(r-1)f(k-1,r)-1.}
4703:
algorithms for finding small sets of elements that contain at least one element from a given family of sets.
1197:
2732:
5869:
The
Sunflower Lemma of Erdős and Rado (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
3498:
6174:
5771:
3920:
307:
108:
The main research question arising in relation to sunflowers is: under what conditions does there exist a
55:
6169:
4866:
4754:-system theorem states that a countable collection of k-sets contains a countably infinite sunflower or
5413:
4869:. It may for example be used as one of the ingredients in a proof showing that it is consistent with
3203:
3129:
2989:
4874:
4863:
4599:. For example, in 1986, Razborov used the sunflower lemma to prove that the Clique language required
1613:
5961:
Kostochka, Alexandr V. (2000), Althöfer, Ingo; Cai, Ning; Dueck, Gunter; Khachatrian, Levon (eds.),
3332:
1421:
6164:
5301:
1927:
3385:
5793:
5645:
5592:
5566:
5546:
5518:
5446:
5335:
5214:
4969:
4942:
4911:
4639:
4602:
1333:
4458:
2343:
372:
5978:
5757:
5714:
5682:
5629:
5536:
4696:
3615:
86:
5890:
4996:
4834:
4811:
4781:
4757:
4737:
4715:
4327:
247:
116:
66:
6109:
6016:
5970:
5820:
5783:
5706:
5674:
5621:
5576:
5528:
5473:
5362:
5261:
5188:
4664:
3440:
3177:
3037:
2963:
2393:
1998:
1766:
981:
906:
880:
854:
694:
587:
5728:
5694:
5641:
5588:
4391:
4364:
4280:
4233:
3988:
3893:
3820:
2466:
2419:
2316:
2209:
2156:
2109:
1971:
1860:
1699:
5724:
5690:
5637:
5584:
4801:
3329:
A month after the release of the first version of their paper, Rao sharpened the bound to
5607:
4511:
4055:
3867:
3794:
3748:
3722:
3656:
3589:
3563:
3103:
2183:
1653:
625:
5388:
5281:
5241:
5194:
5068:
4891:
4645:
4438:
4418:
4307:
4260:
4213:
4081:
4035:
4015:
3847:
3774:
3702:
3682:
3475:
3083:
3063:
2847:
2827:
2807:
2446:
2296:
2276:
2256:
2236:
2136:
1907:
1887:
1746:
1726:
1679:
1528:
1508:
1488:
1468:
1398:
1378:
1313:
1289:
1269:
1159:
1139:
1119:
1099:
1079:
1059:
1036:
1016:
961:
941:
820:
800:
780:
760:
740:
720:
674:
654:
548:
528:
504:
484:
464:
444:
424:
404:
352:
332:
312:
289:
269:
222:
202:
178:
154:
4587:, due to Abott, Hansen, and Sauer. This bound has not been improved in over 50 years.
100:
The naming arises from a visual similarity to the botanical sunflower, arising when a
6153:
6114:
6097:
6021:
6004:
5797:
5747:
5656:
5612:
5596:
5550:
4936:
4852:
4642:
theory at the time. Håstad, Jukna, and Pudlák used it to prove lower bounds on depth-
838:
570:
5649:
5660:
842:
101:
5868:
42:
5974:
1096:
at most once. By adding dummy elements, it suffices to only consider set systems
5846:
5735:
5705:, EATCS Ser. Texts in Theoretical Computer Science, Springer, pp. 210–212,
5603:
5557:
Bell, Tolson; Chueluecha, Suchakree; Warnke, Lutz (2021), "Note on
Sunflowers",
5330:
4859:
1418:
The Erdős-Rado sunflower lemma can be proved directly through induction. First,
1076:
is often assumed to be a set rather than a collection, so any set can appear in
5962:
5580:
3472:. It is equal to the statement that the original sunflower lemma is optimal in
5678:
5610:(1981), "Every large set of equidistant (0,+1,–1)-vectors forms a sunflower",
4855:
4805:
172:
51:
5686:
5633:
5532:
566:
6129:
5710:
5515:
Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of
Computing
112:
sunflower (a sunflower with many sets) in a given collection of sets? The
1307:
5825:
5625:
5499:
4508:
The best existing lower bound for the Erdos-Rado
Sunflower problem for
1156:, so often the sunflower lemma is equivalently phrased as holding for "
649:
620:
15:
5788:
5907:-system". More recently the term "sunflower", possibly introduced by
196:
32:
Collection of sets in which every two sets have the same intersection
5701:
Flum, Jörg; Grohe, Martin (2006), "A Kernelization of
Hitting Set",
5833:
Shanin, N. A. (1946), "A theorem from the general theory of sets",
5571:
5523:
4862:
on the size of a collection of pairwise incompatible elements in a
4638:(superpolynomial) size monotone circuits, a breakthrough result in
848:
Erdos-Rado Delta System
Theorem(corollary of the Sunflower lemma):
6048:
5058:{\displaystyle \langle A_{\alpha }:\alpha <\omega _{2}\rangle }
41:
3080:. The conjecture remains wide open even for fixed low values of
2096:{\displaystyle W_{a}=\{S\setminus \{a\}\mid a\in S,\,S\in W\}.}
1610:
to be a maximal collection of pairwise disjoint sets (that is,
1763:, and a collection of pairwise disjoint sets is a sunflower,
1505:
of these sets make a sunflower. In the general case, suppose
3817:
disjoint copies with the elements added and denote this set
89:
in which all possible distinct pairs of sets share the same
4966:, and if the continuum hypothesis holds, then there is an
1485:
must be a collection of distinct sets of size one, and so
5517:, Association for Computing Machinery, pp. 624–630,
2463:, and so it corresponds to at least one of the sets in a
5180:{\displaystyle f(\alpha )=\sup(A_{\alpha }\cap \alpha )}
5123:{\displaystyle \operatorname {cf} (\alpha )=\omega _{1}}
441:
share the same common subset of elements. An element in
1850:{\displaystyle A=A_{1}\cup A_{2}\cup \cdots \cup A_{t}}
5490:, so by further thinning we may stabilize the kernel.
3612:
sequence of distinct elements is not a sunflower. Let
2273:
to every set. This means that, by our assumption that
5893:
5663:(1960), "Intersection theorems for systems of sets",
5476:
5449:
5416:
5391:
5365:
5338:
5304:
5284:
5264:
5244:
5217:
5197:
5136:
5091:
5071:
5019:
4999:
4972:
4945:
4914:
4894:
4837:
4814:
4784:
4760:
4740:
4718:
4667:
4648:
4605:
4540:
4514:
4461:
4441:
4421:
4394:
4367:
4330:
4310:
4283:
4263:
4236:
4216:
4110:
4084:
4058:
4038:
4018:
3991:
3923:
3896:
3870:
3850:
3823:
3797:
3777:
3751:
3725:
3705:
3685:
3659:
3618:
3592:
3566:
3501:
3478:
3443:
3388:
3335:
3255:
3206:
3180:
3132:
3106:
3086:
3066:
3040:
2992:
2966:
2870:
2850:
2830:
2810:
2735:
2499:
2469:
2449:
2422:
2396:
2346:
2319:
2299:
2279:
2259:
2239:
2212:
2186:
2159:
2139:
2112:
2030:
2001:
1974:
1930:
1910:
1890:
1863:
1798:
1769:
1749:
1729:
1702:
1682:
1656:
1616:
1551:
1531:
1511:
1491:
1471:
1424:
1401:
1381:
1336:
1316:
1292:
1272:
1200:
1162:
1142:
1122:
1102:
1082:
1062:
1039:
1019:
984:
964:
944:
909:
883:
857:
823:
803:
783:
763:
743:
723:
697:
677:
657:
628:
590:
551:
545:
subset, but not others. Note that this intersection,
531:
507:
487:
467:
447:
427:
407:
401:. In other words, a set system or collection of sets
375:
355:
335:
315:
292:
272:
250:
225:
205:
181:
157:
119:
69:
6083:
6035:
5948:
845:, the Delta System Theorem, indicates that it does.
3745:elements and add one element to each set in one of
3437:Erdős and Rado proved the following lower bound on
5899:
5753:Set Theory: An Introduction to Independence Proofs
5482:
5462:
5443:. Using the continuum hypothesis, there are only
5435:
5402:
5377:
5351:
5321:
5290:
5270:
5250:
5230:
5203:
5179:
5122:
5077:
5057:
5005:
4985:
4958:
4927:
4900:
4843:
4820:
4790:
4766:
4746:
4734:which is essentially equivalent to the Erdős-Rado
4724:
4683:
4654:
4630:
4579:
4526:
4497:
4447:
4427:
4407:
4380:
4353:
4316:
4296:
4269:
4249:
4222:
4200:
4090:
4070:
4044:
4024:
4004:
3977:
3909:
3882:
3856:
3836:
3809:
3783:
3763:
3737:
3711:
3691:
3671:
3645:
3604:
3578:
3550:
3484:
3464:
3418:
3374:
3319:
3240:
3192:
3166:
3118:
3092:
3072:
3052:
3026:
2978:
2956:is one of several variations of the conjecture of
2936:
2856:
2836:
2816:
2796:
2718:
2482:
2455:
2435:
2408:
2379:
2332:
2305:
2285:
2265:
2245:
2225:
2198:
2172:
2145:
2125:
2095:
2013:
1987:
1960:
1916:
1896:
1876:
1849:
1781:
1755:
1735:
1715:
1688:
1668:
1642:
1602:
1537:
1517:
1497:
1477:
1457:
1407:
1387:
1367:
1322:
1298:
1278:
1255:
1168:
1148:
1128:
1108:
1088:
1068:
1045:
1025:
1005:
970:
950:
930:
895:
869:
829:
809:
789:
769:
749:
729:
709:
683:
663:
648:, which is defined to be the smallest nonnegative
640:
611:
557:
537:
513:
493:
473:
453:
433:
413:
393:
361:
341:
321:
298:
278:
256:
231:
211:
187:
163:
125:
75:
5925:"Extremal Combinatorics III: Some Basic Theorems"
4595:The sunflower lemma has numerous applications in
4201:{\displaystyle f(a+b,r)\geq (f(a,r)-1)(f(b,r)-1)}
5152:
584:Specifically, researchers analyze the function
1603:{\displaystyle A_{1},A_{2},\ldots ,A_{t}\in W}
6096:Abbott, H.L; Hanson, D; Sauer, N (May 1972).
6003:Abbott, H.L; Hanson, D; Sauer, N (May 1972).
5969:, Boston, MA: Springer US, pp. 143–150,
4257:be two sunflower free families. For each set
3320:{\displaystyle C=O(r^{3}\log(k)\log \log(k))}
8:
5052:
5020:
4580:{\displaystyle 10^{\frac {k}{2}}\leq f(k,3)}
4101:A stronger result is the following theorem:
2087:
2059:
2053:
2044:
817:sets. Though it is not obvious that such an
6098:"Intersection theorems for systems of sets"
6005:"Intersection theorems for systems of sets"
5908:
5666:Journal of the London Mathematical Society
4691:circuits. It has also been applied in the
2206:elements. Furthermore, every sunflower of
6140:
6113:
6102:Journal of Combinatorial Theory, Series A
6020:
6009:Journal of Combinatorial Theory, Series A
5892:
5824:
5787:
5570:
5522:
5475:
5454:
5448:
5421:
5415:
5390:
5364:
5343:
5337:
5303:
5283:
5263:
5243:
5222:
5216:
5196:
5162:
5135:
5114:
5090:
5070:
5046:
5027:
5018:
4998:
4977:
4971:
4950:
4944:
4919:
4913:
4893:
4836:
4813:
4783:
4759:
4739:
4717:
4707:Analogue for infinite collections of sets
4675:
4666:
4647:
4610:
4604:
4545:
4539:
4513:
4490:
4482:
4477:
4471:
4462:
4460:
4440:
4420:
4399:
4393:
4372:
4366:
4346:
4340:
4331:
4329:
4309:
4288:
4282:
4262:
4241:
4235:
4215:
4109:
4083:
4057:
4037:
4017:
4012:is sunflower free since any selection of
3996:
3990:
3970:
3964:
3955:
3947:
3939:
3922:
3901:
3895:
3869:
3849:
3828:
3822:
3796:
3776:
3750:
3724:
3719:be such a set. Take an additional set of
3704:
3684:
3658:
3617:
3591:
3565:
3518:
3500:
3477:
3442:
3387:
3334:
3272:
3254:
3232:
3205:
3179:
3158:
3131:
3105:
3085:
3065:
3039:
3018:
2991:
2965:
2957:
2937:{\displaystyle f(k,r)\leq k(r-1)f(k-1,r)}
2869:
2849:
2829:
2809:
2744:
2736:
2734:
2660:
2652:
2563:
2555:
2547:
2541:
2532:
2520:
2508:
2500:
2498:
2474:
2468:
2448:
2427:
2421:
2395:
2345:
2324:
2318:
2298:
2278:
2258:
2238:
2217:
2211:
2185:
2164:
2158:
2138:
2117:
2111:
2077:
2035:
2029:
2000:
1979:
1973:
1929:
1909:
1889:
1868:
1862:
1841:
1822:
1809:
1797:
1768:
1748:
1728:
1707:
1701:
1681:
1655:
1634:
1621:
1615:
1588:
1569:
1556:
1550:
1530:
1510:
1490:
1470:
1423:
1400:
1380:
1359:
1335:
1315:
1291:
1271:
1244:
1199:
1184:
1161:
1141:
1121:
1101:
1081:
1061:
1038:
1018:
983:
963:
943:
908:
882:
856:
837:must exist, a basic and simple result of
822:
802:
782:
762:
742:
722:
696:
676:
656:
627:
589:
550:
530:
506:
486:
466:
446:
426:
406:
374:
354:
334:
314:
291:
271:
249:
224:
204:
180:
156:
118:
93:. This common intersection is called the
68:
6049:"Quanta Magazine - Illuminating Science"
5887:The original term for this concept was "
1330:of cardinality greater than or equal to
6130:https://mathoverflow.net/a/420288/12176
6128:Lower Bounds for the Sunflower Problem
5919:
5917:
5880:
5852:The sunflower lemma via Shannon entropy
2050:
1256:{\displaystyle f(k,r)\leq k!(r-1)^{k}.}
27:(more unsolved problems in mathematics)
4882:
4361:many sets. Denote this family of sets
3653:denote the size of the largest set of
2797:{\displaystyle |W|\geq k(r-1)f(k-1,r)}
481:or else appears in at most one of the
3551:{\displaystyle (r-1)^{k}\leq f(k,r).}
978:-sets is of cardinality greater than
461:is either found in the common subset
7:
6084:Bell, Chueluecha & Warnke (2021)
4877:does not hold. It was introduced by
3978:{\displaystyle (r-1)|H|\leq |H^{*}|}
6071:
5967:Numbers, Information and Complexity
4591:Applications of the sunflower lemma
1395:contains a sunflower with at least
5911:, has been gradually replacing it.
5894:
5000:
4838:
4815:
4785:
4761:
4741:
4719:
3382:; the current best-known bound is
251:
120:
70:
14:
4858:tool used in proofs to impose an
5963:"Extremal Problems on Δ-Systems"
5436:{\displaystyle A_{i}\subseteq j}
3241:{\displaystyle f(k,r)\leq C^{k}}
3167:{\displaystyle f(k,3)\leq C^{k}}
3027:{\displaystyle f(k,r)\leq C^{k}}
5835:C. R. (Doklady) Acad. Sci. URSS
5703:Parameterized Complexity Theory
2948:Erdős-Rado sunflower conjecture
2153:, except that every element of
1643:{\displaystyle A_{i}\cap A_{j}}
141:Erdős-Rado sunflower conjecture
18:Unsolved problem in mathematics
5806:"Sunflowers: from soil to oil"
5174:
5155:
5146:
5140:
5104:
5098:
4623:
4617:
4574:
4562:
4505:sets which is sunflower free.
4491:
4483:
4478:
4463:
4347:
4332:
4195:
4186:
4174:
4168:
4165:
4156:
4144:
4138:
4132:
4114:
3971:
3956:
3948:
3940:
3936:
3924:
3864:with an element added form an
3640:
3622:
3542:
3530:
3515:
3502:
3459:
3447:
3413:
3398:
3375:{\displaystyle C=O(r\log(rk))}
3369:
3366:
3357:
3345:
3314:
3311:
3305:
3290:
3284:
3265:
3222:
3210:
3148:
3136:
3008:
2996:
2931:
2913:
2907:
2895:
2886:
2874:
2791:
2773:
2767:
2755:
2745:
2737:
2707:
2689:
2683:
2671:
2661:
2653:
2646:
2628:
2622:
2610:
2601:
2592:
2574:
2568:
2564:
2556:
2548:
2533:
2509:
2501:
2368:
2350:
2233:corresponds to a sunflower of
1955:
1943:
1458:{\displaystyle f(1,r)\leq r-1}
1440:
1428:
1356:
1343:
1241:
1228:
1216:
1204:
1000:
988:
925:
913:
671:such that, for any set system
606:
594:
577:Sunflower lemma and conjecture
421:is a sunflower if all sets in
50:In the mathematical fields of
1:
5322:{\displaystyle S'\subseteq S}
1961:{\displaystyle kt\leq k(r-1)}
1033:contains a sunflower of size
6115:10.1016/0097-3165(72)90103-3
6022:10.1016/0097-3165(72)90103-3
5975:10.1007/978-1-4757-6048-4_14
5855:, What's new (personal blog)
4597:theoretical computer science
4455:. This produces a family of
3419:{\displaystyle C=O(r\log k)}
5470:-many countable subsets of
5463:{\displaystyle \omega _{1}}
5352:{\displaystyle \omega _{2}}
5231:{\displaystyle \omega _{2}}
4986:{\displaystyle \omega _{2}}
4959:{\displaystyle \omega _{2}}
4928:{\displaystyle \omega _{2}}
4871:Zermelo–Fraenkel set theory
4631:{\displaystyle n^{\log(n)}}
2416:intersects with one of the
1723:). Because we assumed that
1368:{\displaystyle k!(r-1)^{k}}
569:; a collection of pairwise
195:, that is, a collection of
6191:
5581:10.1016/j.disc.2021.112367
4498:{\displaystyle |F^{*}||F|}
4277:in F, append every set in
3126:; it is not known whether
2380:{\displaystyle f(k-1,r)-1}
938:such that if a set system
37:sunflower (disambiguation)
34:
4701:fixed-parameter tractable
2944:and the theorem follows.
2293:has no sunflower of size
1743:had no sunflower of size
394:{\displaystyle A\cap B=S}
5909:Deza & Frankl (1981)
5770:Rao, Anup (2020-02-25),
4808:contains an uncountable
4693:parameterized complexity
3791:. Take the union of the
3646:{\displaystyle h(k-1,r)}
2960:, p. 86) that for each
2253:, simply by adding back
1650:is the empty set unless
797:contains a sunflower of
717:has cardinality at most
501:elements. No element of
6141:Flum & Grohe (2006)
5900:{\displaystyle \Delta }
5772:"Coding for Sunflowers"
5679:10.1112/jlms/s1-35.1.85
5533:10.1145/3357713.3384234
5258:is constantly equal to
5006:{\displaystyle \Delta }
4844:{\displaystyle \Delta }
4821:{\displaystyle \Delta }
4791:{\displaystyle \Delta }
4767:{\displaystyle \Delta }
4747:{\displaystyle \Delta }
4725:{\displaystyle \Delta }
4354:{\displaystyle |F^{*}|}
2443:'s, it intersects with
1465:, since the set system
1375:of sets of cardinality
1176:-uniform" set systems.
1116:such that every set in
266:) if there is a subset
257:{\displaystyle \Delta }
126:{\displaystyle \Delta }
76:{\displaystyle \Delta }
5929:Combinatorics and more
5901:
5813:Bull. Amer. Math. Soc.
5484:
5483:{\displaystyle \beta }
5464:
5437:
5404:
5379:
5378:{\displaystyle i<j}
5353:
5323:
5292:
5272:
5271:{\displaystyle \beta }
5252:
5232:
5205:
5181:
5124:
5079:
5059:
5007:
4987:
4960:
4929:
4902:
4845:
4822:
4792:
4768:
4748:
4726:
4685:
4684:{\displaystyle AC_{0}}
4656:
4632:
4581:
4528:
4499:
4449:
4429:
4409:
4382:
4355:
4318:
4298:
4271:
4251:
4224:
4202:
4092:
4072:
4046:
4026:
4006:
3979:
3911:
3884:
3858:
3838:
3811:
3785:
3765:
3739:
3713:
3693:
3673:
3647:
3606:
3580:
3552:
3486:
3466:
3465:{\displaystyle f(k,r)}
3433:Sunflower lower bounds
3420:
3376:
3321:
3242:
3194:
3193:{\displaystyle C>0}
3168:
3120:
3094:
3074:
3054:
3053:{\displaystyle C>0}
3028:
2980:
2979:{\displaystyle r>2}
2958:Erdős & Rado (1960
2938:
2858:
2844:set sunflower of size
2838:
2818:
2798:
2720:
2484:
2457:
2437:
2410:
2409:{\displaystyle S\in W}
2381:
2334:
2307:
2287:
2267:
2247:
2227:
2200:
2174:
2147:
2133:is a set system, like
2127:
2097:
2015:
2014:{\displaystyle a\in A}
1989:
1962:
1918:
1898:
1878:
1851:
1783:
1782:{\displaystyle t<r}
1757:
1737:
1717:
1690:
1670:
1644:
1604:
1539:
1525:has no sunflower with
1519:
1499:
1479:
1459:
1409:
1389:
1369:
1324:
1300:
1280:
1257:
1185:Erdős & Rado (1960
1170:
1150:
1130:
1110:
1090:
1070:
1047:
1027:
1007:
1006:{\displaystyle f(k,r)}
972:
952:
932:
931:{\displaystyle f(k,r)}
903:, there is an integer
897:
896:{\displaystyle r>0}
871:
870:{\displaystyle k>0}
831:
811:
791:
771:
751:
731:
711:
710:{\displaystyle S\in W}
685:
665:
642:
613:
612:{\displaystyle f(k,r)}
559:
539:
515:
495:
475:
455:
435:
415:
395:
363:
343:
323:
300:
280:
258:
233:
213:
189:
165:
127:
77:
56:extremal combinatorics
47:
6160:Forcing (mathematics)
6036:Alweiss et al. (2020)
5949:Alweiss et al. (2020)
5902:
5711:10.1007/3-540-29953-X
5485:
5465:
5438:
5405:
5380:
5354:
5324:
5293:
5273:
5253:
5233:
5206:
5182:
5125:
5080:
5060:
5008:
4988:
4961:
4935:-sized collection of
4930:
4903:
4846:
4823:
4793:
4769:
4749:
4727:
4686:
4657:
4633:
4582:
4529:
4500:
4450:
4430:
4410:
4408:{\displaystyle F_{A}}
4383:
4381:{\displaystyle F_{A}}
4356:
4319:
4299:
4297:{\displaystyle F^{*}}
4272:
4252:
4250:{\displaystyle F^{*}}
4225:
4203:
4093:
4073:
4047:
4027:
4007:
4005:{\displaystyle H^{*}}
3980:
3912:
3910:{\displaystyle H^{*}}
3885:
3859:
3839:
3837:{\displaystyle H^{*}}
3812:
3786:
3766:
3740:
3714:
3694:
3674:
3648:
3607:
3581:
3553:
3487:
3467:
3421:
3377:
3322:
3243:
3195:
3169:
3121:
3095:
3075:
3055:
3029:
2981:
2939:
2859:
2839:
2819:
2799:
2721:
2485:
2483:{\displaystyle W_{a}}
2458:
2438:
2436:{\displaystyle A_{i}}
2411:
2382:
2335:
2333:{\displaystyle W_{a}}
2308:
2288:
2268:
2248:
2228:
2226:{\displaystyle W_{a}}
2201:
2175:
2173:{\displaystyle W_{a}}
2148:
2128:
2126:{\displaystyle W_{a}}
2098:
2016:
1990:
1988:{\displaystyle W_{a}}
1963:
1919:
1904:, the cardinality of
1899:
1879:
1877:{\displaystyle A_{i}}
1852:
1784:
1758:
1738:
1718:
1716:{\displaystyle A_{i}}
1696:intersects with some
1691:
1671:
1645:
1605:
1540:
1520:
1500:
1480:
1460:
1410:
1390:
1370:
1325:
1301:
1281:
1258:
1171:
1151:
1131:
1111:
1091:
1071:
1048:
1028:
1008:
973:
953:
933:
898:
872:
832:
812:
792:
772:
752:
732:
712:
686:
666:
643:
614:
560:
540:
516:
496:
476:
456:
436:
416:
396:
364:
344:
324:
301:
281:
259:
234:
214:
190:
166:
128:
78:
45:
5891:
5559:Discrete Mathematics
5474:
5447:
5414:
5389:
5363:
5336:
5302:
5282:
5262:
5242:
5215:
5195:
5134:
5089:
5069:
5017:
4997:
4970:
4943:
4912:
4892:
4875:continuum hypothesis
4835:
4812:
4782:
4758:
4738:
4716:
4665:
4646:
4603:
4538:
4512:
4459:
4439:
4419:
4392:
4388:. Take the union of
4365:
4328:
4308:
4281:
4261:
4234:
4214:
4108:
4082:
4056:
4036:
4016:
3989:
3921:
3894:
3868:
3848:
3821:
3795:
3775:
3749:
3723:
3703:
3683:
3657:
3616:
3590:
3564:
3499:
3476:
3441:
3386:
3333:
3253:
3204:
3178:
3130:
3104:
3084:
3064:
3038:
2990:
2964:
2954:sunflower conjecture
2868:
2848:
2828:
2808:
2733:
2497:
2467:
2447:
2420:
2394:
2344:
2317:
2297:
2277:
2257:
2237:
2210:
2184:
2157:
2137:
2110:
2028:
1999:
1972:
1928:
1908:
1888:
1861:
1796:
1767:
1747:
1727:
1700:
1680:
1654:
1614:
1549:
1545:sets. Then consider
1529:
1509:
1489:
1469:
1422:
1399:
1379:
1334:
1314:
1310:, then a set system
1290:
1270:
1198:
1191:, which states that
1187:, p. 86) proved the
1160:
1140:
1120:
1100:
1080:
1060:
1037:
1017:
982:
962:
942:
907:
881:
855:
821:
801:
781:
761:
741:
721:
695:
691:such that every set
675:
655:
626:
588:
549:
529:
505:
485:
465:
445:
425:
405:
373:
353:
333:
313:
290:
270:
248:
223:
203:
179:
155:
117:
67:
35:For other uses, see
5931:. 28 September 2008
5359:such that whenever
4697:hitting set problem
4527:{\displaystyle r=3}
4071:{\displaystyle r-1}
3883:{\displaystyle r-1}
3810:{\displaystyle r-1}
3771:disjoint copies of
3764:{\displaystyle r-1}
3738:{\displaystyle r-1}
3672:{\displaystyle k-1}
3605:{\displaystyle r-1}
3579:{\displaystyle k=1}
3119:{\displaystyle r=3}
2340:must be bounded by
2199:{\displaystyle k-1}
1676:, and every set in
1669:{\displaystyle i=j}
1056:In the literature,
641:{\displaystyle k,r}
306:such that for each
85:is a collection of
5897:
5804:Rao, Anup (2023),
5626:10.1007/BF02579328
5480:
5460:
5433:
5403:{\displaystyle S'}
5400:
5375:
5349:
5319:
5288:
5268:
5248:
5228:
5201:
5177:
5120:
5075:
5055:
5003:
4983:
4956:
4925:
4898:
4841:
4818:
4800:states that every
4788:
4764:
4744:
4722:
4681:
4652:
4640:circuit complexity
4628:
4577:
4524:
4495:
4445:
4425:
4405:
4378:
4351:
4314:
4294:
4267:
4247:
4220:
4198:
4088:
4068:
4042:
4022:
4002:
3975:
3907:
3880:
3854:
3834:
3807:
3781:
3761:
3735:
3709:
3689:
3669:
3643:
3602:
3576:
3548:
3482:
3462:
3416:
3372:
3317:
3238:
3190:
3164:
3116:
3090:
3070:
3060:depending only on
3050:
3034:for some constant
3024:
2976:
2934:
2854:
2834:
2814:
2794:
2716:
2531:
2480:
2453:
2433:
2406:
2377:
2330:
2303:
2283:
2263:
2243:
2223:
2196:
2170:
2143:
2123:
2093:
2011:
1985:
1958:
1914:
1894:
1874:
1847:
1779:
1753:
1733:
1713:
1686:
1666:
1640:
1600:
1535:
1515:
1495:
1475:
1455:
1405:
1385:
1365:
1320:
1296:
1276:
1253:
1166:
1146:
1126:
1106:
1086:
1066:
1043:
1023:
1003:
968:
948:
928:
893:
867:
827:
807:
787:
767:
747:
727:
707:
681:
661:
638:
609:
555:
535:
521:is shared by just
511:
491:
471:
451:
431:
411:
391:
359:
339:
319:
296:
276:
254:
229:
209:
185:
161:
123:
97:of the sunflower.
73:
48:
5984:978-1-4757-6048-4
5826:10.1090/bull/1777
5789:10.19086/da.11887
5776:Discrete Analysis
5763:978-0-444-85401-8
5756:, North-Holland,
5720:978-3-540-29952-3
5669:, Second Series,
5542:978-1-4503-6979-4
5291:{\displaystyle S}
5251:{\displaystyle f}
5204:{\displaystyle S}
5078:{\displaystyle W}
4901:{\displaystyle W}
4711:A version of the
4655:{\displaystyle 3}
4553:
4448:{\displaystyle F}
4428:{\displaystyle A}
4317:{\displaystyle A}
4270:{\displaystyle A}
4223:{\displaystyle F}
4091:{\displaystyle r}
4045:{\displaystyle r}
4025:{\displaystyle r}
3857:{\displaystyle H}
3784:{\displaystyle H}
3712:{\displaystyle H}
3692:{\displaystyle r}
3485:{\displaystyle r}
3093:{\displaystyle r}
3073:{\displaystyle r}
2857:{\displaystyle k}
2837:{\displaystyle r}
2817:{\displaystyle W}
2516:
2456:{\displaystyle A}
2306:{\displaystyle r}
2286:{\displaystyle W}
2266:{\displaystyle a}
2246:{\displaystyle W}
2146:{\displaystyle W}
1917:{\displaystyle A}
1897:{\displaystyle k}
1756:{\displaystyle r}
1736:{\displaystyle W}
1689:{\displaystyle W}
1538:{\displaystyle r}
1518:{\displaystyle W}
1498:{\displaystyle r}
1478:{\displaystyle W}
1408:{\displaystyle r}
1388:{\displaystyle k}
1323:{\displaystyle W}
1299:{\displaystyle r}
1279:{\displaystyle k}
1169:{\displaystyle k}
1149:{\displaystyle k}
1129:{\displaystyle W}
1109:{\displaystyle W}
1089:{\displaystyle W}
1069:{\displaystyle W}
1046:{\displaystyle r}
1026:{\displaystyle F}
971:{\displaystyle k}
951:{\displaystyle F}
830:{\displaystyle n}
810:{\displaystyle r}
790:{\displaystyle W}
770:{\displaystyle n}
750:{\displaystyle W}
730:{\displaystyle k}
684:{\displaystyle W}
664:{\displaystyle n}
558:{\displaystyle S}
538:{\displaystyle W}
514:{\displaystyle U}
494:{\displaystyle W}
474:{\displaystyle S}
454:{\displaystyle U}
434:{\displaystyle W}
414:{\displaystyle W}
362:{\displaystyle W}
342:{\displaystyle B}
322:{\displaystyle A}
299:{\displaystyle U}
279:{\displaystyle S}
232:{\displaystyle W}
219:. The collection
212:{\displaystyle U}
188:{\displaystyle U}
164:{\displaystyle W}
147:Formal definition
6182:
6144:
6138:
6132:
6126:
6120:
6119:
6117:
6093:
6087:
6081:
6075:
6069:
6063:
6062:
6060:
6059:
6045:
6039:
6033:
6027:
6026:
6024:
6000:
5994:
5993:
5992:
5991:
5958:
5952:
5946:
5940:
5939:
5937:
5936:
5921:
5912:
5906:
5904:
5903:
5898:
5885:
5867:Thiemann, René.
5856:
5842:
5829:
5828:
5810:
5800:
5791:
5766:
5743:
5731:
5697:
5652:
5599:
5574:
5553:
5526:
5489:
5487:
5486:
5481:
5469:
5467:
5466:
5461:
5459:
5458:
5442:
5440:
5439:
5434:
5426:
5425:
5409:
5407:
5406:
5401:
5399:
5384:
5382:
5381:
5376:
5358:
5356:
5355:
5350:
5348:
5347:
5328:
5326:
5325:
5320:
5312:
5297:
5295:
5294:
5289:
5277:
5275:
5274:
5269:
5257:
5255:
5254:
5249:
5237:
5235:
5234:
5229:
5227:
5226:
5210:
5208:
5207:
5202:
5186:
5184:
5183:
5178:
5167:
5166:
5129:
5127:
5126:
5121:
5119:
5118:
5084:
5082:
5081:
5076:
5064:
5062:
5061:
5056:
5051:
5050:
5032:
5031:
5013:-subsystem. Let
5012:
5010:
5009:
5004:
4992:
4990:
4989:
4984:
4982:
4981:
4965:
4963:
4962:
4957:
4955:
4954:
4934:
4932:
4931:
4926:
4924:
4923:
4907:
4905:
4904:
4899:
4850:
4848:
4847:
4842:
4827:
4825:
4824:
4819:
4797:
4795:
4794:
4789:
4773:
4771:
4770:
4765:
4753:
4751:
4750:
4745:
4731:
4729:
4728:
4723:
4690:
4688:
4687:
4682:
4680:
4679:
4661:
4659:
4658:
4653:
4637:
4635:
4634:
4629:
4627:
4626:
4586:
4584:
4583:
4578:
4555:
4554:
4546:
4533:
4531:
4530:
4525:
4504:
4502:
4501:
4496:
4494:
4486:
4481:
4476:
4475:
4466:
4454:
4452:
4451:
4446:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4404:
4403:
4387:
4385:
4384:
4379:
4377:
4376:
4360:
4358:
4357:
4352:
4350:
4345:
4344:
4335:
4323:
4321:
4320:
4315:
4303:
4301:
4300:
4295:
4293:
4292:
4276:
4274:
4273:
4268:
4256:
4254:
4253:
4248:
4246:
4245:
4229:
4227:
4226:
4221:
4207:
4205:
4204:
4199:
4097:
4095:
4094:
4089:
4077:
4075:
4074:
4069:
4051:
4049:
4048:
4043:
4031:
4029:
4028:
4023:
4011:
4009:
4008:
4003:
4001:
4000:
3984:
3982:
3981:
3976:
3974:
3969:
3968:
3959:
3951:
3943:
3916:
3914:
3913:
3908:
3906:
3905:
3889:
3887:
3886:
3881:
3863:
3861:
3860:
3855:
3844:. The copies of
3843:
3841:
3840:
3835:
3833:
3832:
3816:
3814:
3813:
3808:
3790:
3788:
3787:
3782:
3770:
3768:
3767:
3762:
3744:
3742:
3741:
3736:
3718:
3716:
3715:
3710:
3698:
3696:
3695:
3690:
3678:
3676:
3675:
3670:
3652:
3650:
3649:
3644:
3611:
3609:
3608:
3603:
3585:
3583:
3582:
3577:
3557:
3555:
3554:
3549:
3523:
3522:
3491:
3489:
3488:
3483:
3471:
3469:
3468:
3463:
3425:
3423:
3422:
3417:
3381:
3379:
3378:
3373:
3328:
3326:
3324:
3323:
3318:
3277:
3276:
3247:
3245:
3244:
3239:
3237:
3236:
3199:
3197:
3196:
3191:
3173:
3171:
3170:
3165:
3163:
3162:
3125:
3123:
3122:
3117:
3099:
3097:
3096:
3091:
3079:
3077:
3076:
3071:
3059:
3057:
3056:
3051:
3033:
3031:
3030:
3025:
3023:
3022:
2985:
2983:
2982:
2977:
2943:
2941:
2940:
2935:
2863:
2861:
2860:
2855:
2843:
2841:
2840:
2835:
2823:
2821:
2820:
2815:
2803:
2801:
2800:
2795:
2748:
2740:
2725:
2723:
2722:
2717:
2664:
2656:
2567:
2559:
2551:
2546:
2545:
2536:
2530:
2512:
2504:
2489:
2487:
2486:
2481:
2479:
2478:
2462:
2460:
2459:
2454:
2442:
2440:
2439:
2434:
2432:
2431:
2415:
2413:
2412:
2407:
2390:Since every set
2386:
2384:
2383:
2378:
2339:
2337:
2336:
2331:
2329:
2328:
2312:
2310:
2309:
2304:
2292:
2290:
2289:
2284:
2272:
2270:
2269:
2264:
2252:
2250:
2249:
2244:
2232:
2230:
2229:
2224:
2222:
2221:
2205:
2203:
2202:
2197:
2179:
2177:
2176:
2171:
2169:
2168:
2152:
2150:
2149:
2144:
2132:
2130:
2129:
2124:
2122:
2121:
2102:
2100:
2099:
2094:
2040:
2039:
2020:
2018:
2017:
2012:
1994:
1992:
1991:
1986:
1984:
1983:
1967:
1965:
1964:
1959:
1923:
1921:
1920:
1915:
1903:
1901:
1900:
1895:
1884:has cardinality
1883:
1881:
1880:
1875:
1873:
1872:
1856:
1854:
1853:
1848:
1846:
1845:
1827:
1826:
1814:
1813:
1788:
1786:
1785:
1780:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1722:
1720:
1719:
1714:
1712:
1711:
1695:
1693:
1692:
1687:
1675:
1673:
1672:
1667:
1649:
1647:
1646:
1641:
1639:
1638:
1626:
1625:
1609:
1607:
1606:
1601:
1593:
1592:
1574:
1573:
1561:
1560:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1484:
1482:
1481:
1476:
1464:
1462:
1461:
1456:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1364:
1363:
1329:
1327:
1326:
1321:
1305:
1303:
1302:
1297:
1285:
1283:
1282:
1277:
1262:
1260:
1259:
1254:
1249:
1248:
1175:
1173:
1172:
1167:
1155:
1153:
1152:
1147:
1136:has cardinality
1135:
1133:
1132:
1127:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1052:
1050:
1049:
1044:
1032:
1030:
1029:
1024:
1012:
1010:
1009:
1004:
977:
975:
974:
969:
957:
955:
954:
949:
937:
935:
934:
929:
902:
900:
899:
894:
876:
874:
873:
868:
836:
834:
833:
828:
816:
814:
813:
808:
796:
794:
793:
788:
776:
774:
773:
768:
756:
754:
753:
748:
736:
734:
733:
728:
716:
714:
713:
708:
690:
688:
687:
682:
670:
668:
667:
662:
647:
645:
644:
639:
619:for nonnegative
618:
616:
615:
610:
564:
562:
561:
556:
544:
542:
541:
536:
520:
518:
517:
512:
500:
498:
497:
492:
480:
478:
477:
472:
460:
458:
457:
452:
440:
438:
437:
432:
420:
418:
417:
412:
400:
398:
397:
392:
368:
366:
365:
360:
348:
346:
345:
340:
328:
326:
325:
320:
305:
303:
302:
297:
285:
283:
282:
277:
263:
261:
260:
255:
238:
236:
235:
230:
218:
216:
215:
210:
194:
192:
191:
186:
170:
168:
167:
162:
132:
130:
129:
124:
82:
80:
79:
74:
19:
6190:
6189:
6185:
6184:
6183:
6181:
6180:
6179:
6150:
6149:
6148:
6147:
6139:
6135:
6127:
6123:
6095:
6094:
6090:
6082:
6078:
6070:
6066:
6057:
6055:
6053:Quanta Magazine
6047:
6046:
6042:
6034:
6030:
6002:
6001:
5997:
5989:
5987:
5985:
5960:
5959:
5955:
5947:
5943:
5934:
5932:
5923:
5922:
5915:
5889:
5888:
5886:
5882:
5877:
5864:
5859:
5845:
5832:
5808:
5803:
5769:
5764:
5746:
5734:
5721:
5700:
5655:
5602:
5556:
5543:
5512:
5508:
5496:
5472:
5471:
5450:
5445:
5444:
5417:
5412:
5411:
5392:
5387:
5386:
5361:
5360:
5339:
5334:
5333:
5305:
5300:
5299:
5280:
5279:
5260:
5259:
5240:
5239:
5218:
5213:
5212:
5193:
5192:
5158:
5132:
5131:
5110:
5087:
5086:
5067:
5066:
5042:
5023:
5015:
5014:
4995:
4994:
4973:
4968:
4967:
4946:
4941:
4940:
4915:
4910:
4909:
4890:
4889:
4833:
4832:
4810:
4809:
4780:
4779:
4756:
4755:
4736:
4735:
4714:
4713:
4709:
4671:
4663:
4662:
4644:
4643:
4606:
4601:
4600:
4593:
4541:
4536:
4535:
4510:
4509:
4467:
4457:
4456:
4437:
4436:
4417:
4416:
4395:
4390:
4389:
4368:
4363:
4362:
4336:
4326:
4325:
4306:
4305:
4284:
4279:
4278:
4259:
4258:
4237:
4232:
4231:
4212:
4211:
4106:
4105:
4080:
4079:
4054:
4053:
4034:
4033:
4014:
4013:
3992:
3987:
3986:
3960:
3919:
3918:
3917:. We have that,
3897:
3892:
3891:
3866:
3865:
3846:
3845:
3824:
3819:
3818:
3793:
3792:
3773:
3772:
3747:
3746:
3721:
3720:
3701:
3700:
3699:sunflower. Let
3681:
3680:
3655:
3654:
3614:
3613:
3588:
3587:
3562:
3561:
3514:
3497:
3496:
3474:
3473:
3439:
3438:
3435:
3429:
3384:
3383:
3331:
3330:
3268:
3251:
3250:
3249:
3228:
3202:
3201:
3176:
3175:
3154:
3128:
3127:
3102:
3101:
3082:
3081:
3062:
3061:
3036:
3035:
3014:
2988:
2987:
2962:
2961:
2950:
2866:
2865:
2846:
2845:
2826:
2825:
2806:
2805:
2731:
2730:
2537:
2495:
2494:
2470:
2465:
2464:
2445:
2444:
2423:
2418:
2417:
2392:
2391:
2342:
2341:
2320:
2315:
2314:
2295:
2294:
2275:
2274:
2255:
2254:
2235:
2234:
2213:
2208:
2207:
2182:
2181:
2160:
2155:
2154:
2135:
2134:
2113:
2108:
2107:
2031:
2026:
2025:
1997:
1996:
1975:
1970:
1969:
1926:
1925:
1906:
1905:
1886:
1885:
1864:
1859:
1858:
1837:
1818:
1805:
1794:
1793:
1765:
1764:
1745:
1744:
1725:
1724:
1703:
1698:
1697:
1678:
1677:
1652:
1651:
1630:
1617:
1612:
1611:
1584:
1565:
1552:
1547:
1546:
1527:
1526:
1507:
1506:
1487:
1486:
1467:
1466:
1420:
1419:
1397:
1396:
1377:
1376:
1355:
1332:
1331:
1312:
1311:
1288:
1287:
1268:
1267:
1240:
1196:
1195:
1189:sunflower lemma
1182:
1180:Sunflower lemma
1158:
1157:
1138:
1137:
1118:
1117:
1098:
1097:
1078:
1077:
1058:
1057:
1035:
1034:
1015:
1014:
980:
979:
960:
959:
940:
939:
905:
904:
879:
878:
853:
852:
819:
818:
799:
798:
779:
778:
759:
758:
739:
738:
719:
718:
693:
692:
673:
672:
653:
652:
624:
623:
586:
585:
579:
547:
546:
527:
526:
503:
502:
483:
482:
463:
462:
443:
442:
423:
422:
403:
402:
371:
370:
351:
350:
331:
330:
311:
310:
288:
287:
268:
267:
246:
245:
221:
220:
201:
200:
177:
176:
153:
152:
149:
137:sunflower lemma
115:
114:
65:
64:
40:
33:
30:
29:
24:
21:
17:
12:
11:
5:
6188:
6186:
6178:
6177:
6172:
6167:
6162:
6152:
6151:
6146:
6145:
6133:
6121:
6108:(3): 381–389.
6088:
6076:
6064:
6040:
6028:
6015:(3): 381–389.
5995:
5983:
5953:
5941:
5913:
5896:
5879:
5878:
5876:
5873:
5872:
5871:
5863:
5862:External links
5860:
5858:
5857:
5843:
5837:, New Series,
5830:
5801:
5767:
5762:
5748:Kunen, Kenneth
5744:
5732:
5719:
5698:
5653:
5620:(3): 225–231,
5600:
5554:
5541:
5509:
5507:
5504:
5503:
5502:
5495:
5492:
5479:
5457:
5453:
5432:
5429:
5424:
5420:
5398:
5395:
5374:
5371:
5368:
5346:
5342:
5318:
5315:
5311:
5308:
5287:
5267:
5247:
5225:
5221:
5211:stationary in
5200:
5176:
5173:
5170:
5165:
5161:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5117:
5113:
5109:
5106:
5103:
5100:
5097:
5094:
5074:
5054:
5049:
5045:
5041:
5038:
5035:
5030:
5026:
5022:
5002:
4980:
4976:
4953:
4949:
4922:
4918:
4897:
4840:
4817:
4804:collection of
4787:
4763:
4743:
4721:
4708:
4705:
4678:
4674:
4670:
4651:
4625:
4622:
4619:
4616:
4613:
4609:
4592:
4589:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4552:
4549:
4544:
4523:
4520:
4517:
4493:
4489:
4485:
4480:
4474:
4470:
4465:
4444:
4424:
4402:
4398:
4375:
4371:
4349:
4343:
4339:
4334:
4313:
4291:
4287:
4266:
4244:
4240:
4219:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4087:
4067:
4064:
4061:
4041:
4021:
3999:
3995:
3973:
3967:
3963:
3958:
3954:
3950:
3946:
3942:
3938:
3935:
3932:
3929:
3926:
3904:
3900:
3879:
3876:
3873:
3853:
3831:
3827:
3806:
3803:
3800:
3780:
3760:
3757:
3754:
3734:
3731:
3728:
3708:
3688:
3679:-sets with no
3668:
3665:
3662:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3601:
3598:
3595:
3575:
3572:
3569:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3521:
3517:
3513:
3510:
3507:
3504:
3481:
3461:
3458:
3455:
3452:
3449:
3446:
3434:
3431:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3275:
3271:
3267:
3264:
3261:
3258:
3235:
3231:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3189:
3186:
3183:
3161:
3157:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3115:
3112:
3109:
3100:; for example
3089:
3069:
3049:
3046:
3043:
3021:
3017:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2975:
2972:
2969:
2949:
2946:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2853:
2833:
2813:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2747:
2743:
2739:
2727:
2726:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2663:
2659:
2655:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2566:
2562:
2558:
2554:
2550:
2544:
2540:
2535:
2529:
2526:
2523:
2519:
2515:
2511:
2507:
2503:
2477:
2473:
2452:
2430:
2426:
2405:
2402:
2399:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2327:
2323:
2313:, the size of
2302:
2282:
2262:
2242:
2220:
2216:
2195:
2192:
2189:
2167:
2163:
2142:
2120:
2116:
2104:
2103:
2092:
2089:
2086:
2083:
2080:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2038:
2034:
2010:
2007:
2004:
1982:
1978:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1924:is bounded by
1913:
1893:
1871:
1867:
1844:
1840:
1836:
1833:
1830:
1825:
1821:
1817:
1812:
1808:
1804:
1801:
1778:
1775:
1772:
1752:
1732:
1710:
1706:
1685:
1665:
1662:
1659:
1637:
1633:
1629:
1624:
1620:
1599:
1596:
1591:
1587:
1583:
1580:
1577:
1572:
1568:
1564:
1559:
1555:
1534:
1514:
1494:
1474:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1404:
1384:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1319:
1295:
1275:
1264:
1263:
1252:
1247:
1243:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1181:
1178:
1165:
1145:
1125:
1105:
1085:
1065:
1042:
1022:
1002:
999:
996:
993:
990:
987:
967:
947:
927:
924:
921:
918:
915:
912:
892:
889:
886:
866:
863:
860:
826:
806:
786:
766:
757:has more than
746:
726:
706:
703:
700:
680:
660:
637:
634:
631:
608:
605:
602:
599:
596:
593:
578:
575:
554:
534:
510:
490:
470:
450:
430:
410:
390:
387:
384:
381:
378:
358:
338:
318:
295:
275:
253:
228:
208:
184:
160:
148:
145:
122:
72:
31:
25:
22:
16:
13:
10:
9:
6:
4:
3:
2:
6187:
6176:
6175:Combinatorics
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6157:
6155:
6142:
6137:
6134:
6131:
6125:
6122:
6116:
6111:
6107:
6103:
6099:
6092:
6089:
6085:
6080:
6077:
6073:
6068:
6065:
6054:
6050:
6044:
6041:
6037:
6032:
6029:
6023:
6018:
6014:
6010:
6006:
5999:
5996:
5986:
5980:
5976:
5972:
5968:
5964:
5957:
5954:
5950:
5945:
5942:
5930:
5926:
5920:
5918:
5914:
5910:
5884:
5881:
5874:
5870:
5866:
5865:
5861:
5854:
5853:
5848:
5844:
5840:
5836:
5831:
5827:
5822:
5818:
5814:
5807:
5802:
5799:
5795:
5790:
5785:
5781:
5777:
5773:
5768:
5765:
5759:
5755:
5754:
5749:
5745:
5741:
5737:
5733:
5730:
5726:
5722:
5716:
5712:
5708:
5704:
5699:
5696:
5692:
5688:
5684:
5680:
5676:
5672:
5668:
5667:
5662:
5658:
5654:
5651:
5647:
5643:
5639:
5635:
5631:
5627:
5623:
5619:
5615:
5614:
5613:Combinatorica
5609:
5605:
5601:
5598:
5594:
5590:
5586:
5582:
5578:
5573:
5568:
5565:(7): 112367,
5564:
5560:
5555:
5552:
5548:
5544:
5538:
5534:
5530:
5525:
5520:
5516:
5511:
5510:
5505:
5501:
5498:
5497:
5493:
5491:
5477:
5455:
5451:
5430:
5427:
5422:
5418:
5396:
5393:
5372:
5369:
5366:
5344:
5340:
5332:
5316:
5313:
5309:
5306:
5285:
5265:
5245:
5223:
5219:
5198:
5190:
5189:Fodor's lemma
5171:
5168:
5163:
5159:
5149:
5143:
5137:
5115:
5111:
5107:
5101:
5095:
5092:
5072:
5047:
5043:
5039:
5036:
5033:
5028:
5024:
4978:
4974:
4951:
4947:
4938:
4920:
4916:
4895:
4886:
4884:
4880:
4876:
4872:
4868:
4865:
4861:
4857:
4856:set-theoretic
4854:
4853:combinatorial
4829:
4807:
4803:
4799:
4775:
4733:
4706:
4704:
4702:
4698:
4694:
4676:
4672:
4668:
4649:
4641:
4620:
4614:
4611:
4607:
4598:
4590:
4588:
4571:
4568:
4565:
4559:
4556:
4550:
4547:
4542:
4521:
4518:
4515:
4506:
4487:
4472:
4468:
4442:
4422:
4400:
4396:
4373:
4369:
4341:
4337:
4311:
4289:
4285:
4264:
4242:
4238:
4217:
4208:
4192:
4189:
4183:
4180:
4177:
4171:
4162:
4159:
4153:
4150:
4147:
4141:
4135:
4129:
4126:
4123:
4120:
4117:
4111:
4102:
4099:
4085:
4065:
4062:
4059:
4039:
4019:
3997:
3993:
3965:
3961:
3952:
3944:
3933:
3930:
3927:
3902:
3898:
3890:partition of
3877:
3874:
3871:
3851:
3829:
3825:
3804:
3801:
3798:
3778:
3758:
3755:
3752:
3732:
3729:
3726:
3706:
3686:
3666:
3663:
3660:
3637:
3634:
3631:
3628:
3625:
3619:
3599:
3596:
3593:
3573:
3570:
3567:
3558:
3545:
3539:
3536:
3533:
3527:
3524:
3519:
3511:
3508:
3505:
3493:
3479:
3456:
3453:
3450:
3444:
3432:
3430:
3427:
3410:
3407:
3404:
3401:
3395:
3392:
3389:
3363:
3360:
3354:
3351:
3348:
3342:
3339:
3336:
3308:
3302:
3299:
3296:
3293:
3287:
3281:
3278:
3273:
3269:
3262:
3259:
3256:
3233:
3229:
3225:
3219:
3216:
3213:
3207:
3187:
3184:
3181:
3159:
3155:
3151:
3145:
3142:
3139:
3133:
3113:
3110:
3107:
3087:
3067:
3047:
3044:
3041:
3019:
3015:
3011:
3005:
3002:
2999:
2993:
2973:
2970:
2967:
2959:
2955:
2947:
2945:
2928:
2925:
2922:
2919:
2916:
2910:
2904:
2901:
2898:
2892:
2889:
2883:
2880:
2877:
2871:
2864:sets. Hence,
2851:
2831:
2811:
2788:
2785:
2782:
2779:
2776:
2770:
2764:
2761:
2758:
2752:
2749:
2741:
2713:
2710:
2704:
2701:
2698:
2695:
2692:
2686:
2680:
2677:
2674:
2668:
2665:
2657:
2649:
2643:
2640:
2637:
2634:
2631:
2625:
2619:
2616:
2613:
2607:
2604:
2598:
2595:
2589:
2586:
2583:
2580:
2577:
2571:
2560:
2552:
2542:
2538:
2527:
2524:
2521:
2517:
2513:
2505:
2493:
2492:
2491:
2475:
2471:
2450:
2428:
2424:
2403:
2400:
2397:
2388:
2374:
2371:
2365:
2362:
2359:
2356:
2353:
2347:
2325:
2321:
2300:
2280:
2260:
2240:
2218:
2214:
2193:
2190:
2187:
2165:
2161:
2140:
2118:
2114:
2090:
2084:
2081:
2078:
2074:
2071:
2068:
2065:
2062:
2056:
2047:
2041:
2036:
2032:
2024:
2023:
2022:
2008:
2005:
2002:
1980:
1976:
1952:
1949:
1946:
1940:
1937:
1934:
1931:
1911:
1891:
1869:
1865:
1857:. Since each
1842:
1838:
1834:
1831:
1828:
1823:
1819:
1815:
1810:
1806:
1802:
1799:
1790:
1776:
1773:
1770:
1750:
1730:
1708:
1704:
1683:
1663:
1660:
1657:
1635:
1631:
1627:
1622:
1618:
1597:
1594:
1589:
1585:
1581:
1578:
1575:
1570:
1566:
1562:
1557:
1553:
1532:
1512:
1492:
1472:
1452:
1449:
1446:
1443:
1437:
1434:
1431:
1425:
1416:
1402:
1382:
1360:
1352:
1349:
1346:
1340:
1337:
1317:
1309:
1306:are positive
1293:
1273:
1250:
1245:
1237:
1234:
1231:
1225:
1222:
1219:
1213:
1210:
1207:
1201:
1194:
1193:
1192:
1190:
1186:
1179:
1177:
1163:
1143:
1123:
1103:
1083:
1063:
1054:
1040:
1020:
997:
994:
991:
985:
965:
945:
922:
919:
916:
910:
890:
887:
884:
864:
861:
858:
849:
846:
844:
840:
824:
804:
784:
764:
744:
724:
704:
701:
698:
678:
658:
651:
635:
632:
629:
622:
603:
600:
597:
591:
582:
576:
574:
572:
568:
552:
532:
524:
508:
488:
468:
448:
428:
408:
388:
385:
382:
379:
376:
356:
336:
316:
309:
293:
273:
265:
242:
226:
206:
198:
182:
174:
158:
146:
144:
142:
138:
134:
111:
106:
103:
98:
96:
92:
88:
84:
61:
57:
53:
44:
38:
28:
6170:Intersection
6136:
6124:
6105:
6101:
6091:
6079:
6067:
6056:. Retrieved
6052:
6043:
6031:
6012:
6008:
5998:
5988:, retrieved
5966:
5956:
5951:, p. 3.
5944:
5933:. Retrieved
5928:
5883:
5851:
5847:Tao, Terence
5838:
5834:
5819:(1): 29–38,
5816:
5812:
5782:(2): 11887,
5779:
5775:
5752:
5739:
5736:Jech, Thomas
5702:
5673:(1): 85–90,
5670:
5664:
5617:
5611:
5562:
5558:
5514:
4887:
4851:-lemma is a
4830:
4778:
4776:
4712:
4710:
4699:, to design
4594:
4507:
4209:
4103:
4100:
3559:
3494:
3436:
3428:
2953:
2951:
2824:contains an
2728:
2389:
2105:
1791:
1417:
1266:That is, if
1265:
1188:
1183:
1055:
850:
847:
583:
580:
522:
244:
240:
150:
140:
136:
113:
109:
107:
102:Venn diagram
99:
94:
91:intersection
63:
59:
49:
5657:Erdős, Paul
5331:cardinality
4939:subsets of
4860:upper bound
4806:finite sets
4802:uncountable
4324:to produce
4210:Proof. Let
3560:Proof. For
777:sets, then
6165:Set theory
6154:Categories
6072:Rao (2020)
6058:2019-11-10
5990:2022-05-02
5935:2021-12-10
5742:, Springer
5740:Set Theory
5608:Frankl, P.
5572:2009.09327
5524:1908.08483
5506:References
5238:such that
5065:enumerate
2729:Hence, if
369:, we have
173:set system
139:, and the
52:set theory
5895:Δ
5841:: 399–400
5798:202558957
5687:0024-6107
5634:0209-9683
5597:221818818
5551:201314765
5478:β
5452:ω
5428:⊆
5341:ω
5314:⊆
5266:β
5220:ω
5172:α
5169:∩
5164:α
5144:α
5112:ω
5102:α
5096:
5053:⟩
5044:ω
5037:α
5029:α
5021:⟨
5001:Δ
4975:ω
4948:ω
4937:countable
4917:ω
4873:that the
4839:Δ
4828:-system.
4816:Δ
4786:Δ
4774:-system.
4762:Δ
4742:Δ
4720:Δ
4615:
4557:≤
4473:∗
4415:over all
4342:∗
4290:∗
4243:∗
4190:−
4160:−
4136:≥
4104:Theorem.
4063:−
3998:∗
3966:∗
3953:≤
3931:−
3903:∗
3875:−
3830:∗
3802:−
3756:−
3730:−
3664:−
3629:−
3597:−
3586:a set of
3525:≤
3509:−
3495:Theorem.
3408:
3355:
3303:
3297:
3282:
3226:≤
3174:for some
3152:≤
3012:≤
2920:−
2902:−
2890:≤
2780:−
2762:−
2750:≥
2711:−
2696:−
2678:−
2666:≤
2650:−
2635:−
2617:−
2605:≤
2596:−
2581:−
2553:≤
2525:∈
2518:∑
2514:≤
2401:∈
2372:−
2357:−
2191:−
2082:∈
2069:∈
2063:∣
2051:∖
2006:∈
1995:for some
1968:. Define
1950:−
1938:≤
1835:∪
1832:⋯
1829:∪
1816:∪
1628:∩
1595:∈
1579:…
1450:−
1444:≤
1350:−
1235:−
1220:≤
851:For each
702:∈
565:, may be
380:∩
252:Δ
241:sunflower
199:of a set
151:Suppose
121:Δ
71:Δ
60:sunflower
5849:(2020),
5750:(1980),
5738:(2003),
5661:Rado, R.
5650:14043028
5604:Deza, M.
5494:See also
5397:′
5310:′
5298:. Build
1308:integers
621:integers
571:disjoint
308:distinct
5729:2238686
5695:0111692
5642:0637827
5589:4240687
5500:Cap set
5385:are in
5130:, let
5085:. For
4993:-sized
4881: (
4864:forcing
4695:of the
2804:, then
1013:, then
650:integer
525:of the
264:-system
197:subsets
83:-system
5981:
5796:
5760:
5727:
5717:
5693:
5685:
5648:
5640:
5632:
5595:
5587:
5549:
5539:
5191:, fix
5187:. By
4908:is an
4879:Shanin
4798:-lemma
4732:-lemma
4098:sets.
2021:to be
1415:sets.
133:-lemma
95:kernel
5875:Notes
5809:(PDF)
5794:S2CID
5646:S2CID
5593:S2CID
5567:arXiv
5547:S2CID
5519:arXiv
5410:then
4867:poset
2106:Then
839:Erdős
737:, if
567:empty
239:is a
175:over
171:is a
110:large
5979:ISBN
5780:2020
5758:ISBN
5715:ISBN
5683:ISSN
5630:ISSN
5537:ISBN
5370:<
5040:<
4883:1946
4831:The
4777:The
4230:and
3248:for
3185:>
3045:>
2971:>
2952:The
2180:has
1792:Let
1774:<
1286:and
888:>
862:>
843:Rado
841:and
523:some
329:and
243:(or
87:sets
58:, a
54:and
6110:doi
6017:doi
5971:doi
5821:doi
5784:doi
5707:doi
5675:doi
5622:doi
5577:doi
5563:344
5529:doi
5329:of
5278:on
5153:sup
4888:If
4885:).
4612:log
4534:is
4435:in
4304:to
3405:log
3352:log
3300:log
3294:log
3279:log
958:of
349:in
286:of
62:or
6156::
6106:12
6104:.
6100:.
6051:.
6013:12
6011:.
6007:.
5977:,
5965:,
5927:.
5916:^
5839:53
5817:60
5815:,
5811:,
5792:,
5778:,
5774:,
5725:MR
5723:,
5713:,
5691:MR
5689:,
5681:,
5671:35
5659:;
5644:,
5638:MR
5636:,
5628:,
5616:,
5606:;
5591:,
5585:MR
5583:,
5575:,
5561:,
5545:,
5535:,
5527:,
5093:cf
4543:10
3985:.
3492:.
3426:.
2986:,
2714:1.
2490::
2387:.
1789:.
1053:.
877:,
135:,
6143:.
6118:.
6112::
6086:.
6074:.
6061:.
6038:.
6025:.
6019::
5973::
5938:.
5823::
5786::
5709::
5677::
5624::
5618:1
5579::
5569::
5531::
5521::
5456:1
5431:j
5423:i
5419:A
5394:S
5373:j
5367:i
5345:2
5317:S
5307:S
5286:S
5246:f
5224:2
5199:S
5175:)
5160:A
5156:(
5150:=
5147:)
5141:(
5138:f
5116:1
5108:=
5105:)
5099:(
5073:W
5048:2
5034::
5025:A
4979:2
4952:2
4921:2
4896:W
4677:0
4673:C
4669:A
4650:3
4624:)
4621:n
4618:(
4608:n
4575:)
4572:3
4569:,
4566:k
4563:(
4560:f
4551:2
4548:k
4522:3
4519:=
4516:r
4492:|
4488:F
4484:|
4479:|
4469:F
4464:|
4443:F
4423:A
4401:A
4397:F
4374:A
4370:F
4348:|
4338:F
4333:|
4312:A
4286:F
4265:A
4239:F
4218:F
4196:)
4193:1
4187:)
4184:r
4181:,
4178:b
4175:(
4172:f
4169:(
4166:)
4163:1
4157:)
4154:r
4151:,
4148:a
4145:(
4142:f
4139:(
4133:)
4130:r
4127:,
4124:b
4121:+
4118:a
4115:(
4112:f
4086:r
4066:1
4060:r
4040:r
4020:r
3994:H
3972:|
3962:H
3957:|
3949:|
3945:H
3941:|
3937:)
3934:1
3928:r
3925:(
3899:H
3878:1
3872:r
3852:H
3826:H
3805:1
3799:r
3779:H
3759:1
3753:r
3733:1
3727:r
3707:H
3687:r
3667:1
3661:k
3641:)
3638:r
3635:,
3632:1
3626:k
3623:(
3620:h
3600:1
3594:r
3574:1
3571:=
3568:k
3546:.
3543:)
3540:r
3537:,
3534:k
3531:(
3528:f
3520:k
3516:)
3512:1
3506:r
3503:(
3480:r
3460:)
3457:r
3454:,
3451:k
3448:(
3445:f
3414:)
3411:k
3402:r
3399:(
3396:O
3393:=
3390:C
3370:)
3367:)
3364:k
3361:r
3358:(
3349:r
3346:(
3343:O
3340:=
3337:C
3327:.
3315:)
3312:)
3309:k
3306:(
3291:)
3288:k
3285:(
3274:3
3270:r
3266:(
3263:O
3260:=
3257:C
3234:k
3230:C
3223:)
3220:r
3217:,
3214:k
3211:(
3208:f
3188:0
3182:C
3160:k
3156:C
3149:)
3146:3
3143:,
3140:k
3137:(
3134:f
3114:3
3111:=
3108:r
3088:r
3068:r
3048:0
3042:C
3020:k
3016:C
3009:)
3006:r
3003:,
3000:k
2997:(
2994:f
2974:2
2968:r
2932:)
2929:r
2926:,
2923:1
2917:k
2914:(
2911:f
2908:)
2905:1
2899:r
2896:(
2893:k
2887:)
2884:r
2881:,
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2875:(
2872:f
2852:k
2832:r
2812:W
2792:)
2789:r
2786:,
2783:1
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2774:(
2771:f
2768:)
2765:1
2759:r
2756:(
2753:k
2746:|
2742:W
2738:|
2708:)
2705:r
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2699:1
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2690:(
2687:f
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2681:1
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2672:(
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2662:|
2658:A
2654:|
2647:)
2644:r
2641:,
2638:1
2632:k
2629:(
2626:f
2623:)
2620:1
2614:r
2611:(
2608:k
2602:)
2599:1
2593:)
2590:r
2587:,
2584:1
2578:k
2575:(
2572:f
2569:(
2565:|
2561:A
2557:|
2549:|
2543:a
2539:W
2534:|
2528:A
2522:a
2510:|
2506:W
2502:|
2476:a
2472:W
2451:A
2429:i
2425:A
2404:W
2398:S
2375:1
2369:)
2366:r
2363:,
2360:1
2354:k
2351:(
2348:f
2326:a
2322:W
2301:r
2281:W
2261:a
2241:W
2219:a
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2194:1
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2166:a
2162:W
2141:W
2119:a
2115:W
2091:.
2088:}
2085:W
2079:S
2075:,
2072:S
2066:a
2060:}
2057:a
2054:{
2048:S
2045:{
2042:=
2037:a
2033:W
2009:A
2003:a
1981:a
1977:W
1956:)
1953:1
1947:r
1944:(
1941:k
1935:t
1932:k
1912:A
1892:k
1870:i
1866:A
1843:t
1839:A
1824:2
1820:A
1811:1
1807:A
1803:=
1800:A
1777:r
1771:t
1751:r
1731:W
1709:i
1705:A
1684:W
1664:j
1661:=
1658:i
1636:j
1632:A
1623:i
1619:A
1598:W
1590:t
1586:A
1582:,
1576:,
1571:2
1567:A
1563:,
1558:1
1554:A
1533:r
1513:W
1493:r
1473:W
1453:1
1447:r
1441:)
1438:r
1435:,
1432:1
1429:(
1426:f
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1383:k
1361:k
1357:)
1353:1
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1344:(
1341:!
1338:k
1318:W
1294:r
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1251:.
1246:k
1242:)
1238:1
1232:r
1229:(
1226:!
1223:k
1217:)
1214:r
1211:,
1208:k
1205:(
1202:f
1164:k
1144:k
1124:W
1104:W
1084:W
1064:W
1041:r
1021:F
1001:)
998:r
995:,
992:k
989:(
986:f
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946:F
926:)
923:r
920:,
917:k
914:(
911:f
891:0
885:r
865:0
859:k
825:n
805:r
785:W
765:n
745:W
725:k
705:W
699:S
679:W
659:n
636:r
633:,
630:k
607:)
604:r
601:,
598:k
595:(
592:f
553:S
533:W
509:U
489:W
469:S
449:U
429:W
409:W
389:S
386:=
383:B
377:A
357:W
337:B
317:A
294:U
274:S
227:W
207:U
183:U
159:W
39:.
20::
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