54:
115:
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
2735:
5074:
2112:
5196:
5139:
1866:
154:
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.
4217:
1619:
3336:
4596:
2953:
57:
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.
1272:
2813:
3567:
2507:
3994:
5338:
5452:
3257:
3183:
3043:
1659:
3391:
1474:
592:
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.
1977:
5762:
3435:
5479:
5368:
5247:
5002:
4975:
4944:
4647:
1384:
4514:
2396:
410:
3662:
5916:
5419:
5022:
4860:
4837:
4807:
4783:
4763:
4741:
4370:
273:
142:
92:
5499:
5394:
5287:
4700:
3481:
3209:
3069:
2995:
2425:
2030:
1798:
1022:
947:
912:
886:
726:
628:
4424:
4397:
4313:
4266:
4021:
3926:
3853:
2499:
2452:
2349:
2242:
2189:
2142:
2004:
1893:
1732:
4543:
4087:
3899:
3826:
3780:
3754:
3688:
3621:
3595:
3135:
2215:
1685:
657:
5307:
5267:
5220:
5094:
4917:
4671:
4464:
4444:
4333:
4286:
4239:
4107:
4061:
4041:
3873:
3800:
3728:
3708:
3501:
3109:
3089:
2873:
2853:
2833:
2472:
2322:
2302:
2282:
2262:
2162:
1933:
1913:
1772:
1752:
1705:
1554:
1534:
1514:
1494:
1424:
1404:
1339:
1315:
1295:
1185:
1165:
1145:
1125:
1105:
1085:
1062:
1042:
987:
967:
846:
826:
806:
786:
766:
746:
700:
680:
574:
554:
530:
510:
490:
470:
450:
430:
378:
358:
338:
315:
295:
248:
228:
204:
180:
5676:
34:
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?
37:
5993:
5772:
5729:
5551:
5027:
4089:
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
2038:
5816:
4881:
4063:
sets are selected from across multiple sets of the partition, then two must be selected from one partition since there are only
116:
subsets, which encircle the common elements and one or more unique elements, take on the appearance of the petals of a flower.
5144:
5099:
584:
subsets is also a sunflower. Similarly, a collection of sets each containing the same elements is also trivially a sunflower.
6170:
1806:
5935:
4889:
4607:
4118:
5861:
1559:
4043:
sets if in one of the disjoint partitions is sunflower free by assumption of H being sunflower free. Otherwise, if
3263:
47:
5524:
Alweiss, Ryan; Lovett, Shachar; Wu, Kewen; Zhang, Jiapeng (June 2020), "Improved bounds for the sunflower lemma",
4548:
4711:
101:
4703:
2878:
3211:. A 2021 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving that
2730:{\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.}
4714:
algorithms for finding small sets of elements that contain at least one element from a given family of sets.
1208:
2743:
5880:
The
Sunflower Lemma of Erdős and Rado (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
3509:
6185:
5782:
3931:
318:
119:
The main research question arising in relation to sunflowers is: under what conditions does there exist a
66:
6180:
4877:
4765:-system theorem states that a countable collection of k-sets contains a countably infinite sunflower or
5424:
4880:. It may for example be used as one of the ingredients in a proof showing that it is consistent with
3214:
3140:
3000:
4885:
4874:
4610:. For example, in 1986, Razborov used the sunflower lemma to prove that the Clique language required
1624:
5972:
Kostochka, Alexandr V. (2000), Althöfer, Ingo; Cai, Ning; Dueck, Gunter; Khachatrian, Levon (eds.),
3343:
1432:
6175:
5312:
1938:
3396:
5804:
5656:
5603:
5577:
5557:
5529:
5457:
5346:
5225:
4980:
4953:
4922:
4650:
4613:
1344:
4469:
2354:
383:
5989:
5768:
5725:
5693:
5640:
5547:
4707:
3626:
97:
5901:
5007:
4845:
4822:
4792:
4768:
4748:
4726:
4338:
258:
127:
77:
6120:
6027:
5981:
5831:
5794:
5717:
5685:
5632:
5587:
5539:
5484:
5373:
5272:
5199:
4675:
3451:
3188:
3048:
2974:
2404:
2009:
1777:
992:
917:
891:
865:
705:
598:
5739:
5705:
5652:
5599:
4402:
4375:
4291:
4244:
3999:
3904:
3831:
2477:
2430:
2327:
2220:
2167:
2120:
1982:
1871:
1710:
5735:
5701:
5648:
5595:
4812:
3340:
A month after the release of the first version of their paper, Rao sharpened the bound to
5618:
4522:
4066:
3878:
3805:
3759:
3733:
3667:
3600:
3574:
3114:
2194:
1664:
636:
5399:
5292:
5252:
5205:
5079:
4902:
4656:
4449:
4429:
4318:
4271:
4224:
4092:
4046:
4026:
3858:
3785:
3713:
3693:
3486:
3094:
3074:
2858:
2838:
2818:
2457:
2307:
2287:
2267:
2247:
2147:
1918:
1898:
1757:
1737:
1690:
1539:
1519:
1499:
1479:
1409:
1389:
1324:
1300:
1280:
1170:
1150:
1130:
1110:
1090:
1070:
1047:
1027:
972:
952:
831:
811:
791:
771:
751:
731:
685:
665:
559:
539:
515:
495:
475:
455:
435:
415:
363:
343:
323:
300:
280:
233:
213:
189:
165:
4598:, due to Abott, Hansen, and Sauer. This bound has not been improved in over 50 years.
111:
The naming arises from a visual similarity to the botanical sunflower, arising when a
6164:
6125:
6108:
6032:
6015:
5808:
5758:
5667:
5623:
5607:
5561:
4947:
4863:
4653:
theory at the time. Håstad, Jukna, and Pudlák used it to prove lower bounds on depth-
849:
581:
5660:
5671:
853:
112:
5879:
53:
17:
5985:
1107:
at most once. By adding dummy elements, it suffices to only consider set systems
5857:
5746:
5716:, EATCS Ser. Texts in Theoretical Computer Science, Springer, pp. 210–212,
5614:
5568:
Bell, Tolson; Chueluecha, Suchakree; Warnke, Lutz (2021), "Note on
Sunflowers",
5341:
4870:
1429:
The Erdős-Rado sunflower lemma can be proved directly through induction. First,
1087:
is often assumed to be a set rather than a collection, so any set can appear in
5973:
5591:
3483:. It is equal to the statement that the original sunflower lemma is optimal in
5689:
5621:(1981), "Every large set of equidistant (0,+1,–1)-vectors forms a sunflower",
4866:
4816:
183:
62:
5697:
5644:
5543:
577:
6140:
5721:
5526:
Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of
Computing
123:
sunflower (a sunflower with many sets) in a given collection of sets? The
1318:
5836:
5636:
5510:
4519:
The best existing lower bound for the Erdos-Rado
Sunflower problem for
1167:, so often the sunflower lemma is equivalently phrased as holding for "
660:
631:
26:
5799:
5918:-system". More recently the term "sunflower", possibly introduced by
207:
43:
Collection of sets in which every two sets have the same intersection
5712:
Flum, Jörg; Grohe, Martin (2006), "A Kernelization of
Hitting Set",
5844:
Shanin, N. A. (1946), "A theorem from the general theory of sets",
5582:
5534:
4873:
on the size of a collection of pairwise incompatible elements in a
4649:(superpolynomial) size monotone circuits, a breakthrough result in
859:
Erdos-Rado Delta System
Theorem(corollary of the Sunflower lemma):
6059:
5069:{\displaystyle \langle A_{\alpha }:\alpha <\omega _{2}\rangle }
52:
3091:. The conjecture remains wide open even for fixed low values of
2107:{\displaystyle W_{a}=\{S\setminus \{a\}\mid a\in S,\,S\in W\}.}
1621:
to be a maximal collection of pairwise disjoint sets (that is,
1774:, and a collection of pairwise disjoint sets is a sunflower,
1516:
of these sets make a sunflower. In the general case, suppose
3828:
disjoint copies with the elements added and denote this set
100:
in which all possible distinct pairs of sets share the same
4977:, and if the continuum hypothesis holds, then there is an
1496:
must be a collection of distinct sets of size one, and so
5528:, Association for Computing Machinery, pp. 624–630,
2474:, and so it corresponds to at least one of the sets in a
5191:{\displaystyle f(\alpha )=\sup(A_{\alpha }\cap \alpha )}
5134:{\displaystyle \operatorname {cf} (\alpha )=\omega _{1}}
452:
share the same common subset of elements. An element in
1861:{\displaystyle A=A_{1}\cup A_{2}\cup \cdots \cup A_{t}}
5501:, so by further thinning we may stabilize the kernel.
3623:
sequence of distinct elements is not a sunflower. Let
2284:
to every set. This means that, by our assumption that
5904:
5674:(1960), "Intersection theorems for systems of sets",
5487:
5460:
5427:
5402:
5376:
5349:
5315:
5295:
5275:
5255:
5228:
5208:
5147:
5102:
5082:
5030:
5010:
4983:
4956:
4925:
4905:
4848:
4825:
4795:
4771:
4751:
4729:
4678:
4659:
4616:
4551:
4525:
4472:
4452:
4432:
4405:
4378:
4341:
4321:
4294:
4274:
4247:
4227:
4121:
4095:
4069:
4049:
4029:
4002:
3934:
3907:
3881:
3861:
3834:
3808:
3788:
3762:
3736:
3716:
3696:
3670:
3629:
3603:
3577:
3512:
3489:
3454:
3399:
3346:
3266:
3217:
3191:
3143:
3117:
3097:
3077:
3051:
3003:
2977:
2881:
2861:
2841:
2821:
2746:
2510:
2480:
2460:
2433:
2407:
2357:
2330:
2310:
2290:
2270:
2250:
2223:
2197:
2170:
2150:
2123:
2041:
2012:
1985:
1941:
1921:
1901:
1874:
1809:
1780:
1760:
1740:
1713:
1693:
1667:
1627:
1562:
1542:
1522:
1502:
1482:
1435:
1412:
1392:
1347:
1327:
1303:
1283:
1211:
1173:
1153:
1133:
1113:
1093:
1073:
1050:
1030:
995:
975:
955:
920:
894:
868:
834:
814:
794:
774:
754:
734:
708:
688:
668:
639:
601:
562:
556:
subset, but not others. Note that this intersection,
542:
518:
498:
478:
458:
438:
418:
412:. In other words, a set system or collection of sets
386:
366:
346:
326:
303:
283:
261:
236:
216:
192:
168:
130:
80:
6094:
6046:
5959:
856:, the Delta System Theorem, indicates that it does.
3756:elements and add one element to each set in one of
3448:Erdős and Rado proved the following lower bound on
5910:
5764:Set Theory: An Introduction to Independence Proofs
5493:
5473:
5454:. Using the continuum hypothesis, there are only
5446:
5413:
5388:
5362:
5332:
5301:
5281:
5261:
5241:
5214:
5190:
5133:
5088:
5068:
5016:
4996:
4969:
4938:
4911:
4854:
4831:
4801:
4777:
4757:
4745:which is essentially equivalent to the Erdős-Rado
4735:
4694:
4665:
4641:
4590:
4537:
4508:
4458:
4438:
4418:
4391:
4364:
4327:
4307:
4280:
4260:
4233:
4211:
4101:
4081:
4055:
4035:
4015:
3988:
3920:
3893:
3867:
3847:
3820:
3794:
3774:
3748:
3722:
3702:
3682:
3656:
3615:
3589:
3561:
3495:
3475:
3429:
3385:
3330:
3251:
3203:
3177:
3129:
3103:
3083:
3063:
3037:
2989:
2967:is one of several variations of the conjecture of
2947:
2867:
2847:
2827:
2807:
2729:
2493:
2466:
2446:
2419:
2390:
2343:
2316:
2296:
2276:
2256:
2236:
2209:
2183:
2156:
2136:
2106:
2024:
1998:
1971:
1927:
1907:
1887:
1860:
1792:
1766:
1746:
1726:
1699:
1679:
1653:
1613:
1548:
1528:
1508:
1488:
1468:
1418:
1398:
1378:
1333:
1309:
1289:
1266:
1179:
1159:
1139:
1119:
1099:
1079:
1056:
1036:
1016:
981:
961:
941:
906:
880:
840:
820:
800:
780:
760:
740:
720:
694:
674:
659:, which is defined to be the smallest nonnegative
651:
622:
568:
548:
524:
504:
484:
464:
444:
424:
404:
372:
352:
332:
309:
289:
267:
242:
222:
198:
174:
136:
86:
5936:"Extremal Combinatorics III: Some Basic Theorems"
4606:The sunflower lemma has numerous applications in
4212:{\displaystyle f(a+b,r)\geq (f(a,r)-1)(f(b,r)-1)}
5163:
595:Specifically, researchers analyze the function
1614:{\displaystyle A_{1},A_{2},\ldots ,A_{t}\in W}
6107:Abbott, H.L; Hanson, D; Sauer, N (May 1972).
6014:Abbott, H.L; Hanson, D; Sauer, N (May 1972).
5980:, Boston, MA: Springer US, pp. 143–150,
4268:be two sunflower free families. For each set
3331:{\displaystyle C=O(r^{3}\log(k)\log \log(k))}
8:
5063:
5031:
4591:{\displaystyle 10^{\frac {k}{2}}\leq f(k,3)}
4112:A stronger result is the following theorem:
2098:
2070:
2064:
2055:
828:sets. Though it is not obvious that such an
6109:"Intersection theorems for systems of sets"
6016:"Intersection theorems for systems of sets"
5919:
5677:Journal of the London Mathematical Society
4702:circuits. It has also been applied in the
2217:elements. Furthermore, every sunflower of
6151:
6124:
6113:Journal of Combinatorial Theory, Series A
6031:
6020:Journal of Combinatorial Theory, Series A
5903:
5835:
5798:
5581:
5533:
5486:
5465:
5459:
5432:
5426:
5401:
5375:
5354:
5348:
5314:
5294:
5274:
5254:
5233:
5227:
5207:
5173:
5146:
5125:
5101:
5081:
5057:
5038:
5029:
5009:
4988:
4982:
4961:
4955:
4930:
4924:
4904:
4847:
4824:
4794:
4770:
4750:
4728:
4718:Analogue for infinite collections of sets
4686:
4677:
4658:
4621:
4615:
4556:
4550:
4524:
4501:
4493:
4488:
4482:
4473:
4471:
4451:
4431:
4410:
4404:
4383:
4377:
4357:
4351:
4342:
4340:
4320:
4299:
4293:
4273:
4252:
4246:
4226:
4120:
4094:
4068:
4048:
4028:
4023:is sunflower free since any selection of
4007:
4001:
3981:
3975:
3966:
3958:
3950:
3933:
3912:
3906:
3880:
3860:
3839:
3833:
3807:
3787:
3761:
3735:
3730:be such a set. Take an additional set of
3715:
3695:
3669:
3628:
3602:
3576:
3529:
3511:
3488:
3453:
3398:
3345:
3283:
3265:
3243:
3216:
3190:
3169:
3142:
3116:
3096:
3076:
3050:
3029:
3002:
2976:
2968:
2948:{\displaystyle f(k,r)\leq k(r-1)f(k-1,r)}
2880:
2860:
2840:
2820:
2755:
2747:
2745:
2671:
2663:
2574:
2566:
2558:
2552:
2543:
2531:
2519:
2511:
2509:
2485:
2479:
2459:
2438:
2432:
2406:
2356:
2335:
2329:
2309:
2289:
2269:
2249:
2228:
2222:
2196:
2175:
2169:
2149:
2128:
2122:
2088:
2046:
2040:
2011:
1990:
1984:
1940:
1920:
1900:
1879:
1873:
1852:
1833:
1820:
1808:
1779:
1759:
1739:
1718:
1712:
1692:
1666:
1645:
1632:
1626:
1599:
1580:
1567:
1561:
1541:
1521:
1501:
1481:
1434:
1411:
1391:
1370:
1346:
1326:
1302:
1282:
1255:
1210:
1195:
1172:
1152:
1132:
1112:
1092:
1072:
1049:
1029:
994:
974:
954:
919:
893:
867:
848:must exist, a basic and simple result of
833:
813:
793:
773:
753:
733:
707:
687:
667:
638:
600:
561:
541:
517:
497:
477:
457:
437:
417:
385:
365:
345:
325:
302:
282:
260:
235:
215:
191:
167:
129:
104:. This common intersection is called the
79:
6060:"Quanta Magazine - Illuminating Science"
5898:The original term for this concept was "
1341:of cardinality greater than or equal to
6141:https://mathoverflow.net/a/420288/12176
6139:Lower Bounds for the Sunflower Problem
5930:
5928:
5891:
5863:The sunflower lemma via Shannon entropy
2061:
1267:{\displaystyle f(k,r)\leq k!(r-1)^{k}.}
38:(more unsolved problems in mathematics)
4893:
4372:many sets. Denote this family of sets
3664:denote the size of the largest set of
2808:{\displaystyle |W|\geq k(r-1)f(k-1,r)}
492:or else appears in at most one of the
3562:{\displaystyle (r-1)^{k}\leq f(k,r).}
989:-sets is of cardinality greater than
472:is either found in the common subset
7:
6095:Bell, Chueluecha & Warnke (2021)
4888:does not hold. It was introduced by
3989:{\displaystyle (r-1)|H|\leq |H^{*}|}
6082:
5978:Numbers, Information and Complexity
4602:Applications of the sunflower lemma
1406:contains a sunflower with at least
5922:, has been gradually replacing it.
5905:
5011:
4849:
4826:
4796:
4772:
4752:
4730:
3393:; the current best-known bound is
262:
131:
81:
25:
4869:tool used in proofs to impose an
5974:"Extremal Problems on Δ-Systems"
5447:{\displaystyle A_{i}\subseteq j}
3252:{\displaystyle f(k,r)\leq C^{k}}
3178:{\displaystyle f(k,3)\leq C^{k}}
3038:{\displaystyle f(k,r)\leq C^{k}}
5846:C. R. (Doklady) Acad. Sci. URSS
5714:Parameterized Complexity Theory
2959:Erdős-Rado sunflower conjecture
2164:, except that every element of
1654:{\displaystyle A_{i}\cap A_{j}}
152:Erdős-Rado sunflower conjecture
29:Unsolved problem in mathematics
5817:"Sunflowers: from soil to oil"
5185:
5166:
5157:
5151:
5115:
5109:
4634:
4628:
4585:
4573:
4516:sets which is sunflower free.
4502:
4494:
4489:
4474:
4358:
4343:
4206:
4197:
4185:
4179:
4176:
4167:
4155:
4149:
4143:
4125:
3982:
3967:
3959:
3951:
3947:
3935:
3875:with an element added form an
3651:
3633:
3553:
3541:
3526:
3513:
3470:
3458:
3424:
3409:
3386:{\displaystyle C=O(r\log(rk))}
3380:
3377:
3368:
3356:
3325:
3322:
3316:
3301:
3295:
3276:
3233:
3221:
3159:
3147:
3019:
3007:
2942:
2924:
2918:
2906:
2897:
2885:
2802:
2784:
2778:
2766:
2756:
2748:
2718:
2700:
2694:
2682:
2672:
2664:
2657:
2639:
2633:
2621:
2612:
2603:
2585:
2579:
2575:
2567:
2559:
2544:
2520:
2512:
2379:
2361:
2244:corresponds to a sunflower of
1966:
1954:
1469:{\displaystyle f(1,r)\leq r-1}
1451:
1439:
1367:
1354:
1252:
1239:
1227:
1215:
1011:
999:
936:
924:
682:such that, for any set system
617:
605:
588:Sunflower lemma and conjecture
432:is a sunflower if all sets in
61:In the mathematical fields of
1:
5333:{\displaystyle S'\subseteq S}
1972:{\displaystyle kt\leq k(r-1)}
1044:contains a sunflower of size
6126:10.1016/0097-3165(72)90103-3
6033:10.1016/0097-3165(72)90103-3
5986:10.1007/978-1-4757-6048-4_14
5866:, What's new (personal blog)
4608:theoretical computer science
4466:. This produces a family of
3430:{\displaystyle C=O(r\log k)}
5481:-many countable subsets of
5474:{\displaystyle \omega _{1}}
5363:{\displaystyle \omega _{2}}
5242:{\displaystyle \omega _{2}}
4997:{\displaystyle \omega _{2}}
4970:{\displaystyle \omega _{2}}
4939:{\displaystyle \omega _{2}}
4882:Zermelo–Fraenkel set theory
4642:{\displaystyle n^{\log(n)}}
2427:intersects with one of the
1734:). Because we assumed that
1379:{\displaystyle k!(r-1)^{k}}
580:; a collection of pairwise
206:, that is, a collection of
6202:
5592:10.1016/j.disc.2021.112367
4509:{\displaystyle |F^{*}||F|}
4288:in F, append every set in
3137:; it is not known whether
2391:{\displaystyle f(k-1,r)-1}
949:such that if a set system
48:sunflower (disambiguation)
45:
4712:fixed-parameter tractable
2955:and the theorem follows.
2304:has no sunflower of size
1754:had no sunflower of size
405:{\displaystyle A\cap B=S}
5920:Deza & Frankl (1981)
5781:Rao, Anup (2020-02-25),
4819:contains an uncountable
4704:parameterized complexity
3802:. Take the union of the
3657:{\displaystyle h(k-1,r)}
2971:, p. 86) that for each
2264:, simply by adding back
1661:is the empty set unless
808:contains a sunflower of
728:has cardinality at most
512:elements. No element of
6152:Flum & Grohe (2006)
5911:{\displaystyle \Delta }
5783:"Coding for Sunflowers"
5690:10.1112/jlms/s1-35.1.85
5544:10.1145/3357713.3384234
5269:is constantly equal to
5017:{\displaystyle \Delta }
4855:{\displaystyle \Delta }
4832:{\displaystyle \Delta }
4802:{\displaystyle \Delta }
4778:{\displaystyle \Delta }
4758:{\displaystyle \Delta }
4736:{\displaystyle \Delta }
4365:{\displaystyle |F^{*}|}
2454:'s, it intersects with
1476:, since the set system
1386:of sets of cardinality
1187:-uniform" set systems.
1127:such that every set in
277:) if there is a subset
268:{\displaystyle \Delta }
137:{\displaystyle \Delta }
87:{\displaystyle \Delta }
5940:Combinatorics and more
5912:
5824:Bull. Amer. Math. Soc.
5495:
5494:{\displaystyle \beta }
5475:
5448:
5415:
5390:
5389:{\displaystyle i<j}
5364:
5334:
5303:
5283:
5282:{\displaystyle \beta }
5263:
5243:
5216:
5192:
5135:
5090:
5070:
5018:
4998:
4971:
4940:
4913:
4856:
4833:
4803:
4779:
4759:
4737:
4696:
4695:{\displaystyle AC_{0}}
4667:
4643:
4592:
4539:
4510:
4460:
4440:
4420:
4393:
4366:
4329:
4309:
4282:
4262:
4235:
4213:
4103:
4083:
4057:
4037:
4017:
3990:
3922:
3895:
3869:
3849:
3822:
3796:
3776:
3750:
3724:
3704:
3684:
3658:
3617:
3591:
3563:
3497:
3477:
3476:{\displaystyle f(k,r)}
3444:Sunflower lower bounds
3431:
3387:
3332:
3253:
3205:
3204:{\displaystyle C>0}
3179:
3131:
3105:
3085:
3065:
3064:{\displaystyle C>0}
3039:
2991:
2990:{\displaystyle r>2}
2969:Erdős & Rado (1960
2949:
2869:
2855:set sunflower of size
2849:
2829:
2809:
2731:
2495:
2468:
2448:
2421:
2420:{\displaystyle S\in W}
2392:
2345:
2318:
2298:
2278:
2258:
2238:
2211:
2185:
2158:
2144:is a set system, like
2138:
2108:
2026:
2025:{\displaystyle a\in A}
2000:
1973:
1929:
1909:
1889:
1862:
1794:
1793:{\displaystyle t<r}
1768:
1748:
1728:
1701:
1681:
1655:
1615:
1550:
1536:has no sunflower with
1530:
1510:
1490:
1470:
1420:
1400:
1380:
1335:
1311:
1291:
1268:
1196:Erdős & Rado (1960
1181:
1161:
1141:
1121:
1101:
1081:
1058:
1038:
1018:
1017:{\displaystyle f(k,r)}
983:
963:
943:
942:{\displaystyle f(k,r)}
914:, there is an integer
908:
907:{\displaystyle r>0}
882:
881:{\displaystyle k>0}
842:
822:
802:
782:
762:
742:
722:
721:{\displaystyle S\in W}
696:
676:
653:
624:
623:{\displaystyle f(k,r)}
570:
550:
526:
506:
486:
466:
446:
426:
406:
374:
354:
334:
311:
291:
269:
244:
224:
200:
176:
138:
88:
67:extremal combinatorics
58:
6171:Forcing (mathematics)
6047:Alweiss et al. (2020)
5960:Alweiss et al. (2020)
5913:
5722:10.1007/3-540-29953-X
5496:
5476:
5449:
5416:
5391:
5365:
5335:
5304:
5284:
5264:
5244:
5217:
5193:
5136:
5091:
5071:
5019:
4999:
4972:
4946:-sized collection of
4941:
4914:
4857:
4834:
4804:
4780:
4760:
4738:
4697:
4668:
4644:
4593:
4540:
4511:
4461:
4441:
4421:
4419:{\displaystyle F_{A}}
4394:
4392:{\displaystyle F_{A}}
4367:
4330:
4310:
4308:{\displaystyle F^{*}}
4283:
4263:
4261:{\displaystyle F^{*}}
4236:
4214:
4104:
4084:
4058:
4038:
4018:
4016:{\displaystyle H^{*}}
3991:
3923:
3921:{\displaystyle H^{*}}
3896:
3870:
3850:
3848:{\displaystyle H^{*}}
3823:
3797:
3777:
3751:
3725:
3705:
3685:
3659:
3618:
3592:
3564:
3498:
3478:
3432:
3388:
3333:
3254:
3206:
3180:
3132:
3106:
3086:
3066:
3040:
2992:
2950:
2870:
2850:
2830:
2810:
2732:
2496:
2494:{\displaystyle W_{a}}
2469:
2449:
2447:{\displaystyle A_{i}}
2422:
2393:
2346:
2344:{\displaystyle W_{a}}
2319:
2299:
2279:
2259:
2239:
2237:{\displaystyle W_{a}}
2212:
2186:
2184:{\displaystyle W_{a}}
2159:
2139:
2137:{\displaystyle W_{a}}
2109:
2027:
2001:
1999:{\displaystyle W_{a}}
1974:
1930:
1915:, the cardinality of
1910:
1890:
1888:{\displaystyle A_{i}}
1863:
1795:
1769:
1749:
1729:
1727:{\displaystyle A_{i}}
1707:intersects with some
1702:
1682:
1656:
1616:
1551:
1531:
1511:
1491:
1471:
1421:
1401:
1381:
1336:
1312:
1292:
1269:
1182:
1162:
1142:
1122:
1102:
1082:
1059:
1039:
1019:
984:
964:
944:
909:
883:
843:
823:
803:
783:
763:
743:
723:
697:
677:
654:
625:
571:
551:
527:
507:
487:
467:
447:
427:
407:
375:
355:
335:
312:
292:
270:
245:
225:
201:
177:
139:
89:
56:
5902:
5570:Discrete Mathematics
5485:
5458:
5425:
5400:
5374:
5347:
5313:
5293:
5273:
5253:
5226:
5206:
5145:
5100:
5080:
5028:
5008:
4981:
4954:
4923:
4903:
4886:continuum hypothesis
4846:
4823:
4793:
4769:
4749:
4727:
4676:
4657:
4614:
4549:
4523:
4470:
4450:
4430:
4403:
4399:. Take the union of
4376:
4339:
4319:
4292:
4272:
4245:
4225:
4119:
4093:
4067:
4047:
4027:
4000:
3932:
3905:
3879:
3859:
3832:
3806:
3786:
3760:
3734:
3714:
3694:
3668:
3627:
3601:
3575:
3510:
3487:
3452:
3397:
3344:
3264:
3215:
3189:
3141:
3115:
3095:
3075:
3049:
3001:
2975:
2965:sunflower conjecture
2879:
2859:
2839:
2819:
2744:
2508:
2478:
2458:
2431:
2405:
2355:
2328:
2308:
2288:
2268:
2248:
2221:
2195:
2168:
2148:
2121:
2039:
2010:
1983:
1939:
1919:
1899:
1872:
1807:
1778:
1758:
1738:
1711:
1691:
1665:
1625:
1560:
1556:sets. Then consider
1540:
1520:
1500:
1480:
1433:
1410:
1390:
1345:
1325:
1321:, then a set system
1301:
1281:
1209:
1202:, which states that
1198:, p. 86) proved the
1171:
1151:
1131:
1111:
1091:
1071:
1048:
1028:
993:
973:
953:
918:
892:
866:
832:
812:
792:
772:
752:
732:
706:
702:such that every set
686:
666:
637:
599:
560:
540:
516:
496:
476:
456:
436:
416:
384:
364:
344:
324:
301:
281:
259:
234:
214:
190:
166:
128:
78:
46:For other uses, see
5942:. 28 September 2008
5370:such that whenever
4708:hitting set problem
4538:{\displaystyle r=3}
4082:{\displaystyle r-1}
3894:{\displaystyle r-1}
3821:{\displaystyle r-1}
3782:disjoint copies of
3775:{\displaystyle r-1}
3749:{\displaystyle r-1}
3683:{\displaystyle k-1}
3616:{\displaystyle r-1}
3590:{\displaystyle k=1}
3130:{\displaystyle r=3}
2351:must be bounded by
2210:{\displaystyle k-1}
1687:, and every set in
1680:{\displaystyle i=j}
1067:In the literature,
652:{\displaystyle k,r}
317:such that for each
96:is a collection of
5908:
5815:Rao, Anup (2023),
5637:10.1007/BF02579328
5491:
5471:
5444:
5414:{\displaystyle S'}
5411:
5386:
5360:
5330:
5299:
5279:
5259:
5239:
5212:
5188:
5131:
5086:
5066:
5014:
4994:
4967:
4936:
4909:
4852:
4829:
4811:states that every
4799:
4775:
4755:
4733:
4692:
4663:
4651:circuit complexity
4639:
4588:
4535:
4506:
4456:
4436:
4416:
4389:
4362:
4325:
4305:
4278:
4258:
4231:
4209:
4099:
4079:
4053:
4033:
4013:
3986:
3918:
3891:
3865:
3845:
3818:
3792:
3772:
3746:
3720:
3700:
3680:
3654:
3613:
3587:
3559:
3493:
3473:
3427:
3383:
3328:
3249:
3201:
3175:
3127:
3101:
3081:
3071:depending only on
3061:
3045:for some constant
3035:
2987:
2945:
2865:
2845:
2825:
2805:
2727:
2542:
2491:
2464:
2444:
2417:
2388:
2341:
2314:
2294:
2274:
2254:
2234:
2207:
2181:
2154:
2134:
2104:
2022:
1996:
1969:
1925:
1905:
1885:
1858:
1790:
1764:
1744:
1724:
1697:
1677:
1651:
1611:
1546:
1526:
1506:
1486:
1466:
1416:
1396:
1376:
1331:
1307:
1287:
1264:
1177:
1157:
1137:
1117:
1097:
1077:
1054:
1034:
1014:
979:
959:
939:
904:
878:
838:
818:
798:
778:
758:
738:
718:
692:
672:
649:
620:
566:
546:
532:is shared by just
522:
502:
482:
462:
442:
422:
402:
370:
350:
330:
307:
287:
265:
240:
220:
196:
172:
134:
108:of the sunflower.
84:
59:
18:Delta-system lemma
5995:978-1-4757-6048-4
5837:10.1090/bull/1777
5800:10.19086/da.11887
5787:Discrete Analysis
5774:978-0-444-85401-8
5767:, North-Holland,
5731:978-3-540-29952-3
5680:, Second Series,
5553:978-1-4503-6979-4
5302:{\displaystyle S}
5262:{\displaystyle f}
5215:{\displaystyle S}
5089:{\displaystyle W}
4912:{\displaystyle W}
4722:A version of the
4666:{\displaystyle 3}
4564:
4459:{\displaystyle F}
4439:{\displaystyle A}
4328:{\displaystyle A}
4281:{\displaystyle A}
4234:{\displaystyle F}
4102:{\displaystyle r}
4056:{\displaystyle r}
4036:{\displaystyle r}
3868:{\displaystyle H}
3795:{\displaystyle H}
3723:{\displaystyle H}
3703:{\displaystyle r}
3496:{\displaystyle r}
3104:{\displaystyle r}
3084:{\displaystyle r}
2868:{\displaystyle k}
2848:{\displaystyle r}
2828:{\displaystyle W}
2527:
2467:{\displaystyle A}
2317:{\displaystyle r}
2297:{\displaystyle W}
2277:{\displaystyle a}
2257:{\displaystyle W}
2157:{\displaystyle W}
1928:{\displaystyle A}
1908:{\displaystyle k}
1767:{\displaystyle r}
1747:{\displaystyle W}
1700:{\displaystyle W}
1549:{\displaystyle r}
1529:{\displaystyle W}
1509:{\displaystyle r}
1489:{\displaystyle W}
1419:{\displaystyle r}
1399:{\displaystyle k}
1334:{\displaystyle W}
1310:{\displaystyle r}
1290:{\displaystyle k}
1180:{\displaystyle k}
1160:{\displaystyle k}
1140:{\displaystyle W}
1120:{\displaystyle W}
1100:{\displaystyle W}
1080:{\displaystyle W}
1057:{\displaystyle r}
1037:{\displaystyle F}
982:{\displaystyle k}
962:{\displaystyle F}
841:{\displaystyle n}
821:{\displaystyle r}
801:{\displaystyle W}
781:{\displaystyle n}
761:{\displaystyle W}
741:{\displaystyle k}
695:{\displaystyle W}
675:{\displaystyle n}
569:{\displaystyle S}
549:{\displaystyle W}
525:{\displaystyle U}
505:{\displaystyle W}
485:{\displaystyle S}
465:{\displaystyle U}
445:{\displaystyle W}
425:{\displaystyle W}
373:{\displaystyle W}
353:{\displaystyle B}
333:{\displaystyle A}
310:{\displaystyle U}
290:{\displaystyle S}
243:{\displaystyle W}
230:. The collection
223:{\displaystyle U}
199:{\displaystyle U}
175:{\displaystyle W}
158:Formal definition
16:(Redirected from
6193:
6155:
6149:
6143:
6137:
6131:
6130:
6128:
6104:
6098:
6092:
6086:
6080:
6074:
6073:
6071:
6070:
6056:
6050:
6044:
6038:
6037:
6035:
6011:
6005:
6004:
6003:
6002:
5969:
5963:
5957:
5951:
5950:
5948:
5947:
5932:
5923:
5917:
5915:
5914:
5909:
5896:
5878:Thiemann, René.
5867:
5853:
5840:
5839:
5821:
5811:
5802:
5777:
5754:
5742:
5708:
5663:
5610:
5585:
5564:
5537:
5500:
5498:
5497:
5492:
5480:
5478:
5477:
5472:
5470:
5469:
5453:
5451:
5450:
5445:
5437:
5436:
5420:
5418:
5417:
5412:
5410:
5395:
5393:
5392:
5387:
5369:
5367:
5366:
5361:
5359:
5358:
5339:
5337:
5336:
5331:
5323:
5308:
5306:
5305:
5300:
5288:
5286:
5285:
5280:
5268:
5266:
5265:
5260:
5248:
5246:
5245:
5240:
5238:
5237:
5221:
5219:
5218:
5213:
5197:
5195:
5194:
5189:
5178:
5177:
5140:
5138:
5137:
5132:
5130:
5129:
5095:
5093:
5092:
5087:
5075:
5073:
5072:
5067:
5062:
5061:
5043:
5042:
5024:-subsystem. Let
5023:
5021:
5020:
5015:
5003:
5001:
5000:
4995:
4993:
4992:
4976:
4974:
4973:
4968:
4966:
4965:
4945:
4943:
4942:
4937:
4935:
4934:
4918:
4916:
4915:
4910:
4861:
4859:
4858:
4853:
4838:
4836:
4835:
4830:
4808:
4806:
4805:
4800:
4784:
4782:
4781:
4776:
4764:
4762:
4761:
4756:
4742:
4740:
4739:
4734:
4701:
4699:
4698:
4693:
4691:
4690:
4672:
4670:
4669:
4664:
4648:
4646:
4645:
4640:
4638:
4637:
4597:
4595:
4594:
4589:
4566:
4565:
4557:
4544:
4542:
4541:
4536:
4515:
4513:
4512:
4507:
4505:
4497:
4492:
4487:
4486:
4477:
4465:
4463:
4462:
4457:
4445:
4443:
4442:
4437:
4425:
4423:
4422:
4417:
4415:
4414:
4398:
4396:
4395:
4390:
4388:
4387:
4371:
4369:
4368:
4363:
4361:
4356:
4355:
4346:
4334:
4332:
4331:
4326:
4314:
4312:
4311:
4306:
4304:
4303:
4287:
4285:
4284:
4279:
4267:
4265:
4264:
4259:
4257:
4256:
4240:
4238:
4237:
4232:
4218:
4216:
4215:
4210:
4108:
4106:
4105:
4100:
4088:
4086:
4085:
4080:
4062:
4060:
4059:
4054:
4042:
4040:
4039:
4034:
4022:
4020:
4019:
4014:
4012:
4011:
3995:
3993:
3992:
3987:
3985:
3980:
3979:
3970:
3962:
3954:
3927:
3925:
3924:
3919:
3917:
3916:
3900:
3898:
3897:
3892:
3874:
3872:
3871:
3866:
3855:. The copies of
3854:
3852:
3851:
3846:
3844:
3843:
3827:
3825:
3824:
3819:
3801:
3799:
3798:
3793:
3781:
3779:
3778:
3773:
3755:
3753:
3752:
3747:
3729:
3727:
3726:
3721:
3709:
3707:
3706:
3701:
3689:
3687:
3686:
3681:
3663:
3661:
3660:
3655:
3622:
3620:
3619:
3614:
3596:
3594:
3593:
3588:
3568:
3566:
3565:
3560:
3534:
3533:
3502:
3500:
3499:
3494:
3482:
3480:
3479:
3474:
3436:
3434:
3433:
3428:
3392:
3390:
3389:
3384:
3339:
3337:
3335:
3334:
3329:
3288:
3287:
3258:
3256:
3255:
3250:
3248:
3247:
3210:
3208:
3207:
3202:
3184:
3182:
3181:
3176:
3174:
3173:
3136:
3134:
3133:
3128:
3110:
3108:
3107:
3102:
3090:
3088:
3087:
3082:
3070:
3068:
3067:
3062:
3044:
3042:
3041:
3036:
3034:
3033:
2996:
2994:
2993:
2988:
2954:
2952:
2951:
2946:
2874:
2872:
2871:
2866:
2854:
2852:
2851:
2846:
2834:
2832:
2831:
2826:
2814:
2812:
2811:
2806:
2759:
2751:
2736:
2734:
2733:
2728:
2675:
2667:
2578:
2570:
2562:
2557:
2556:
2547:
2541:
2523:
2515:
2500:
2498:
2497:
2492:
2490:
2489:
2473:
2471:
2470:
2465:
2453:
2451:
2450:
2445:
2443:
2442:
2426:
2424:
2423:
2418:
2401:Since every set
2397:
2395:
2394:
2389:
2350:
2348:
2347:
2342:
2340:
2339:
2323:
2321:
2320:
2315:
2303:
2301:
2300:
2295:
2283:
2281:
2280:
2275:
2263:
2261:
2260:
2255:
2243:
2241:
2240:
2235:
2233:
2232:
2216:
2214:
2213:
2208:
2190:
2188:
2187:
2182:
2180:
2179:
2163:
2161:
2160:
2155:
2143:
2141:
2140:
2135:
2133:
2132:
2113:
2111:
2110:
2105:
2051:
2050:
2031:
2029:
2028:
2023:
2005:
2003:
2002:
1997:
1995:
1994:
1978:
1976:
1975:
1970:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1895:has cardinality
1894:
1892:
1891:
1886:
1884:
1883:
1867:
1865:
1864:
1859:
1857:
1856:
1838:
1837:
1825:
1824:
1799:
1797:
1796:
1791:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1733:
1731:
1730:
1725:
1723:
1722:
1706:
1704:
1703:
1698:
1686:
1684:
1683:
1678:
1660:
1658:
1657:
1652:
1650:
1649:
1637:
1636:
1620:
1618:
1617:
1612:
1604:
1603:
1585:
1584:
1572:
1571:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1515:
1513:
1512:
1507:
1495:
1493:
1492:
1487:
1475:
1473:
1472:
1467:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1375:
1374:
1340:
1338:
1337:
1332:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1273:
1271:
1270:
1265:
1260:
1259:
1186:
1184:
1183:
1178:
1166:
1164:
1163:
1158:
1147:has cardinality
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1063:
1061:
1060:
1055:
1043:
1041:
1040:
1035:
1023:
1021:
1020:
1015:
988:
986:
985:
980:
968:
966:
965:
960:
948:
946:
945:
940:
913:
911:
910:
905:
887:
885:
884:
879:
847:
845:
844:
839:
827:
825:
824:
819:
807:
805:
804:
799:
787:
785:
784:
779:
767:
765:
764:
759:
747:
745:
744:
739:
727:
725:
724:
719:
701:
699:
698:
693:
681:
679:
678:
673:
658:
656:
655:
650:
630:for nonnegative
629:
627:
626:
621:
575:
573:
572:
567:
555:
553:
552:
547:
531:
529:
528:
523:
511:
509:
508:
503:
491:
489:
488:
483:
471:
469:
468:
463:
451:
449:
448:
443:
431:
429:
428:
423:
411:
409:
408:
403:
379:
377:
376:
371:
359:
357:
356:
351:
339:
337:
336:
331:
316:
314:
313:
308:
296:
294:
293:
288:
274:
272:
271:
266:
249:
247:
246:
241:
229:
227:
226:
221:
205:
203:
202:
197:
181:
179:
178:
173:
143:
141:
140:
135:
93:
91:
90:
85:
30:
21:
6201:
6200:
6196:
6195:
6194:
6192:
6191:
6190:
6161:
6160:
6159:
6158:
6150:
6146:
6138:
6134:
6106:
6105:
6101:
6093:
6089:
6081:
6077:
6068:
6066:
6064:Quanta Magazine
6058:
6057:
6053:
6045:
6041:
6013:
6012:
6008:
6000:
5998:
5996:
5971:
5970:
5966:
5958:
5954:
5945:
5943:
5934:
5933:
5926:
5900:
5899:
5897:
5893:
5888:
5875:
5870:
5856:
5843:
5819:
5814:
5780:
5775:
5757:
5745:
5732:
5711:
5666:
5613:
5567:
5554:
5523:
5519:
5507:
5483:
5482:
5461:
5456:
5455:
5428:
5423:
5422:
5403:
5398:
5397:
5372:
5371:
5350:
5345:
5344:
5316:
5311:
5310:
5291:
5290:
5271:
5270:
5251:
5250:
5229:
5224:
5223:
5204:
5203:
5169:
5143:
5142:
5121:
5098:
5097:
5078:
5077:
5053:
5034:
5026:
5025:
5006:
5005:
4984:
4979:
4978:
4957:
4952:
4951:
4926:
4921:
4920:
4901:
4900:
4844:
4843:
4821:
4820:
4791:
4790:
4767:
4766:
4747:
4746:
4725:
4724:
4720:
4682:
4674:
4673:
4655:
4654:
4617:
4612:
4611:
4604:
4552:
4547:
4546:
4521:
4520:
4478:
4468:
4467:
4448:
4447:
4428:
4427:
4406:
4401:
4400:
4379:
4374:
4373:
4347:
4337:
4336:
4317:
4316:
4295:
4290:
4289:
4270:
4269:
4248:
4243:
4242:
4223:
4222:
4117:
4116:
4091:
4090:
4065:
4064:
4045:
4044:
4025:
4024:
4003:
3998:
3997:
3971:
3930:
3929:
3928:. We have that,
3908:
3903:
3902:
3877:
3876:
3857:
3856:
3835:
3830:
3829:
3804:
3803:
3784:
3783:
3758:
3757:
3732:
3731:
3712:
3711:
3710:sunflower. Let
3692:
3691:
3666:
3665:
3625:
3624:
3599:
3598:
3573:
3572:
3525:
3508:
3507:
3485:
3484:
3450:
3449:
3446:
3440:
3395:
3394:
3342:
3341:
3279:
3262:
3261:
3260:
3239:
3213:
3212:
3187:
3186:
3165:
3139:
3138:
3113:
3112:
3093:
3092:
3073:
3072:
3047:
3046:
3025:
2999:
2998:
2973:
2972:
2961:
2877:
2876:
2857:
2856:
2837:
2836:
2817:
2816:
2742:
2741:
2548:
2506:
2505:
2481:
2476:
2475:
2456:
2455:
2434:
2429:
2428:
2403:
2402:
2353:
2352:
2331:
2326:
2325:
2306:
2305:
2286:
2285:
2266:
2265:
2246:
2245:
2224:
2219:
2218:
2193:
2192:
2171:
2166:
2165:
2146:
2145:
2124:
2119:
2118:
2042:
2037:
2036:
2008:
2007:
1986:
1981:
1980:
1937:
1936:
1917:
1916:
1897:
1896:
1875:
1870:
1869:
1848:
1829:
1816:
1805:
1804:
1776:
1775:
1756:
1755:
1736:
1735:
1714:
1709:
1708:
1689:
1688:
1663:
1662:
1641:
1628:
1623:
1622:
1595:
1576:
1563:
1558:
1557:
1538:
1537:
1518:
1517:
1498:
1497:
1478:
1477:
1431:
1430:
1408:
1407:
1388:
1387:
1366:
1343:
1342:
1323:
1322:
1299:
1298:
1279:
1278:
1251:
1207:
1206:
1200:sunflower lemma
1193:
1191:Sunflower lemma
1169:
1168:
1149:
1148:
1129:
1128:
1109:
1108:
1089:
1088:
1069:
1068:
1046:
1045:
1026:
1025:
991:
990:
971:
970:
951:
950:
916:
915:
890:
889:
864:
863:
830:
829:
810:
809:
790:
789:
770:
769:
750:
749:
730:
729:
704:
703:
684:
683:
664:
663:
635:
634:
597:
596:
590:
558:
557:
538:
537:
514:
513:
494:
493:
474:
473:
454:
453:
434:
433:
414:
413:
382:
381:
362:
361:
342:
341:
322:
321:
299:
298:
279:
278:
257:
256:
232:
231:
212:
211:
188:
187:
164:
163:
160:
148:sunflower lemma
126:
125:
76:
75:
51:
44:
41:
40:
35:
32:
28:
23:
22:
15:
12:
11:
5:
6199:
6197:
6189:
6188:
6183:
6178:
6173:
6163:
6162:
6157:
6156:
6144:
6132:
6119:(3): 381–389.
6099:
6087:
6075:
6051:
6039:
6026:(3): 381–389.
6006:
5994:
5964:
5952:
5924:
5907:
5890:
5889:
5887:
5884:
5883:
5882:
5874:
5873:External links
5871:
5869:
5868:
5854:
5848:, New Series,
5841:
5812:
5778:
5773:
5759:Kunen, Kenneth
5755:
5743:
5730:
5709:
5664:
5631:(3): 225–231,
5611:
5565:
5552:
5520:
5518:
5515:
5514:
5513:
5506:
5503:
5490:
5468:
5464:
5443:
5440:
5435:
5431:
5409:
5406:
5385:
5382:
5379:
5357:
5353:
5329:
5326:
5322:
5319:
5298:
5278:
5258:
5236:
5232:
5222:stationary in
5211:
5187:
5184:
5181:
5176:
5172:
5168:
5165:
5162:
5159:
5156:
5153:
5150:
5128:
5124:
5120:
5117:
5114:
5111:
5108:
5105:
5085:
5065:
5060:
5056:
5052:
5049:
5046:
5041:
5037:
5033:
5013:
4991:
4987:
4964:
4960:
4933:
4929:
4908:
4851:
4828:
4815:collection of
4798:
4774:
4754:
4732:
4719:
4716:
4689:
4685:
4681:
4662:
4636:
4633:
4630:
4627:
4624:
4620:
4603:
4600:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4563:
4560:
4555:
4534:
4531:
4528:
4504:
4500:
4496:
4491:
4485:
4481:
4476:
4455:
4435:
4413:
4409:
4386:
4382:
4360:
4354:
4350:
4345:
4324:
4302:
4298:
4277:
4255:
4251:
4230:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4098:
4078:
4075:
4072:
4052:
4032:
4010:
4006:
3984:
3978:
3974:
3969:
3965:
3961:
3957:
3953:
3949:
3946:
3943:
3940:
3937:
3915:
3911:
3890:
3887:
3884:
3864:
3842:
3838:
3817:
3814:
3811:
3791:
3771:
3768:
3765:
3745:
3742:
3739:
3719:
3699:
3690:-sets with no
3679:
3676:
3673:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3612:
3609:
3606:
3586:
3583:
3580:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3532:
3528:
3524:
3521:
3518:
3515:
3492:
3472:
3469:
3466:
3463:
3460:
3457:
3445:
3442:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3286:
3282:
3278:
3275:
3272:
3269:
3246:
3242:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3200:
3197:
3194:
3172:
3168:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3126:
3123:
3120:
3111:; for example
3100:
3080:
3060:
3057:
3054:
3032:
3028:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
2986:
2983:
2980:
2960:
2957:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2864:
2844:
2824:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2758:
2754:
2750:
2738:
2737:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2674:
2670:
2666:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2577:
2573:
2569:
2565:
2561:
2555:
2551:
2546:
2540:
2537:
2534:
2530:
2526:
2522:
2518:
2514:
2488:
2484:
2463:
2441:
2437:
2416:
2413:
2410:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2338:
2334:
2324:, the size of
2313:
2293:
2273:
2253:
2231:
2227:
2206:
2203:
2200:
2178:
2174:
2153:
2131:
2127:
2115:
2114:
2103:
2100:
2097:
2094:
2091:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2049:
2045:
2021:
2018:
2015:
1993:
1989:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1935:is bounded by
1924:
1904:
1882:
1878:
1855:
1851:
1847:
1844:
1841:
1836:
1832:
1828:
1823:
1819:
1815:
1812:
1789:
1786:
1783:
1763:
1743:
1721:
1717:
1696:
1676:
1673:
1670:
1648:
1644:
1640:
1635:
1631:
1610:
1607:
1602:
1598:
1594:
1591:
1588:
1583:
1579:
1575:
1570:
1566:
1545:
1525:
1505:
1485:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1415:
1395:
1373:
1369:
1365:
1362:
1359:
1356:
1353:
1350:
1330:
1306:
1286:
1275:
1274:
1263:
1258:
1254:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1192:
1189:
1176:
1156:
1136:
1116:
1096:
1076:
1053:
1033:
1013:
1010:
1007:
1004:
1001:
998:
978:
958:
938:
935:
932:
929:
926:
923:
903:
900:
897:
877:
874:
871:
837:
817:
797:
777:
768:has more than
757:
737:
717:
714:
711:
691:
671:
648:
645:
642:
619:
616:
613:
610:
607:
604:
589:
586:
565:
545:
521:
501:
481:
461:
441:
421:
401:
398:
395:
392:
389:
369:
349:
329:
306:
286:
264:
239:
219:
195:
171:
159:
156:
133:
83:
42:
36:
33:
27:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6198:
6187:
6186:Combinatorics
6184:
6182:
6179:
6177:
6174:
6172:
6169:
6168:
6166:
6153:
6148:
6145:
6142:
6136:
6133:
6127:
6122:
6118:
6114:
6110:
6103:
6100:
6096:
6091:
6088:
6084:
6079:
6076:
6065:
6061:
6055:
6052:
6048:
6043:
6040:
6034:
6029:
6025:
6021:
6017:
6010:
6007:
5997:
5991:
5987:
5983:
5979:
5975:
5968:
5965:
5961:
5956:
5953:
5941:
5937:
5931:
5929:
5925:
5921:
5895:
5892:
5885:
5881:
5877:
5876:
5872:
5865:
5864:
5859:
5855:
5851:
5847:
5842:
5838:
5833:
5829:
5825:
5818:
5813:
5810:
5806:
5801:
5796:
5792:
5788:
5784:
5779:
5776:
5770:
5766:
5765:
5760:
5756:
5752:
5748:
5744:
5741:
5737:
5733:
5727:
5723:
5719:
5715:
5710:
5707:
5703:
5699:
5695:
5691:
5687:
5683:
5679:
5678:
5673:
5669:
5665:
5662:
5658:
5654:
5650:
5646:
5642:
5638:
5634:
5630:
5626:
5625:
5624:Combinatorica
5620:
5616:
5612:
5609:
5605:
5601:
5597:
5593:
5589:
5584:
5579:
5576:(7): 112367,
5575:
5571:
5566:
5563:
5559:
5555:
5549:
5545:
5541:
5536:
5531:
5527:
5522:
5521:
5516:
5512:
5509:
5508:
5504:
5502:
5488:
5466:
5462:
5441:
5438:
5433:
5429:
5407:
5404:
5383:
5380:
5377:
5355:
5351:
5343:
5327:
5324:
5320:
5317:
5296:
5276:
5256:
5234:
5230:
5209:
5201:
5200:Fodor's lemma
5182:
5179:
5174:
5170:
5160:
5154:
5148:
5126:
5122:
5118:
5112:
5106:
5103:
5083:
5058:
5054:
5050:
5047:
5044:
5039:
5035:
4989:
4985:
4962:
4958:
4949:
4931:
4927:
4906:
4897:
4895:
4891:
4887:
4883:
4879:
4876:
4872:
4868:
4867:set-theoretic
4865:
4864:combinatorial
4840:
4818:
4814:
4810:
4786:
4744:
4717:
4715:
4713:
4709:
4705:
4687:
4683:
4679:
4660:
4652:
4631:
4625:
4622:
4618:
4609:
4601:
4599:
4582:
4579:
4576:
4570:
4567:
4561:
4558:
4553:
4532:
4529:
4526:
4517:
4498:
4483:
4479:
4453:
4433:
4411:
4407:
4384:
4380:
4352:
4348:
4322:
4300:
4296:
4275:
4253:
4249:
4228:
4219:
4203:
4200:
4194:
4191:
4188:
4182:
4173:
4170:
4164:
4161:
4158:
4152:
4146:
4140:
4137:
4134:
4131:
4128:
4122:
4113:
4110:
4096:
4076:
4073:
4070:
4050:
4030:
4008:
4004:
3976:
3972:
3963:
3955:
3944:
3941:
3938:
3913:
3909:
3901:partition of
3888:
3885:
3882:
3862:
3840:
3836:
3815:
3812:
3809:
3789:
3769:
3766:
3763:
3743:
3740:
3737:
3717:
3697:
3677:
3674:
3671:
3648:
3645:
3642:
3639:
3636:
3630:
3610:
3607:
3604:
3584:
3581:
3578:
3569:
3556:
3550:
3547:
3544:
3538:
3535:
3530:
3522:
3519:
3516:
3504:
3490:
3467:
3464:
3461:
3455:
3443:
3441:
3438:
3421:
3418:
3415:
3412:
3406:
3403:
3400:
3374:
3371:
3365:
3362:
3359:
3353:
3350:
3347:
3319:
3313:
3310:
3307:
3304:
3298:
3292:
3289:
3284:
3280:
3273:
3270:
3267:
3244:
3240:
3236:
3230:
3227:
3224:
3218:
3198:
3195:
3192:
3170:
3166:
3162:
3156:
3153:
3150:
3144:
3124:
3121:
3118:
3098:
3078:
3058:
3055:
3052:
3030:
3026:
3022:
3016:
3013:
3010:
3004:
2984:
2981:
2978:
2970:
2966:
2958:
2956:
2939:
2936:
2933:
2930:
2927:
2921:
2915:
2912:
2909:
2903:
2900:
2894:
2891:
2888:
2882:
2875:sets. Hence,
2862:
2842:
2822:
2799:
2796:
2793:
2790:
2787:
2781:
2775:
2772:
2769:
2763:
2760:
2752:
2724:
2721:
2715:
2712:
2709:
2706:
2703:
2697:
2691:
2688:
2685:
2679:
2676:
2668:
2660:
2654:
2651:
2648:
2645:
2642:
2636:
2630:
2627:
2624:
2618:
2615:
2609:
2606:
2600:
2597:
2594:
2591:
2588:
2582:
2571:
2563:
2553:
2549:
2538:
2535:
2532:
2528:
2524:
2516:
2504:
2503:
2502:
2486:
2482:
2461:
2439:
2435:
2414:
2411:
2408:
2399:
2385:
2382:
2376:
2373:
2370:
2367:
2364:
2358:
2336:
2332:
2311:
2291:
2271:
2251:
2229:
2225:
2204:
2201:
2198:
2176:
2172:
2151:
2129:
2125:
2101:
2095:
2092:
2089:
2085:
2082:
2079:
2076:
2073:
2067:
2058:
2052:
2047:
2043:
2035:
2034:
2033:
2019:
2016:
2013:
1991:
1987:
1963:
1960:
1957:
1951:
1948:
1945:
1942:
1922:
1902:
1880:
1876:
1868:. Since each
1853:
1849:
1845:
1842:
1839:
1834:
1830:
1826:
1821:
1817:
1813:
1810:
1801:
1787:
1784:
1781:
1761:
1741:
1719:
1715:
1694:
1674:
1671:
1668:
1646:
1642:
1638:
1633:
1629:
1608:
1605:
1600:
1596:
1592:
1589:
1586:
1581:
1577:
1573:
1568:
1564:
1543:
1523:
1503:
1483:
1463:
1460:
1457:
1454:
1448:
1445:
1442:
1436:
1427:
1413:
1393:
1371:
1363:
1360:
1357:
1351:
1348:
1328:
1320:
1317:are positive
1304:
1284:
1261:
1256:
1248:
1245:
1242:
1236:
1233:
1230:
1224:
1221:
1218:
1212:
1205:
1204:
1203:
1201:
1197:
1190:
1188:
1174:
1154:
1134:
1114:
1094:
1074:
1065:
1051:
1031:
1008:
1005:
1002:
996:
976:
956:
933:
930:
927:
921:
901:
898:
895:
875:
872:
869:
860:
857:
855:
851:
835:
815:
795:
775:
755:
735:
715:
712:
709:
689:
669:
662:
646:
643:
640:
633:
614:
611:
608:
602:
593:
587:
585:
583:
579:
563:
543:
535:
519:
499:
479:
459:
439:
419:
399:
396:
393:
390:
387:
367:
347:
327:
320:
304:
284:
276:
253:
237:
217:
209:
193:
185:
169:
157:
155:
153:
149:
145:
122:
117:
114:
109:
107:
103:
99:
95:
72:
68:
64:
55:
49:
39:
19:
6181:Intersection
6147:
6135:
6116:
6112:
6102:
6090:
6078:
6067:. Retrieved
6063:
6054:
6042:
6023:
6019:
6009:
5999:, retrieved
5977:
5967:
5962:, p. 3.
5955:
5944:. Retrieved
5939:
5894:
5862:
5858:Tao, Terence
5849:
5845:
5830:(1): 29–38,
5827:
5823:
5793:(2): 11887,
5790:
5786:
5763:
5750:
5747:Jech, Thomas
5713:
5684:(1): 85–90,
5681:
5675:
5628:
5622:
5573:
5569:
5525:
4898:
4862:-lemma is a
4841:
4789:
4787:
4723:
4721:
4710:, to design
4605:
4518:
4220:
4114:
4111:
3570:
3505:
3447:
3439:
2964:
2962:
2835:contains an
2739:
2400:
2116:
1802:
1428:
1277:That is, if
1276:
1199:
1194:
1066:
861:
858:
594:
591:
533:
255:
251:
161:
151:
147:
124:
120:
118:
113:Venn diagram
110:
105:
102:intersection
74:
70:
60:
5668:Erdős, Paul
5342:cardinality
4950:subsets of
4871:upper bound
4817:finite sets
4813:uncountable
4335:to produce
4221:Proof. Let
3571:Proof. For
788:sets, then
6176:Set theory
6165:Categories
6083:Rao (2020)
6069:2019-11-10
6001:2022-05-02
5946:2021-12-10
5753:, Springer
5751:Set Theory
5619:Frankl, P.
5583:2009.09327
5535:1908.08483
5517:References
5249:such that
5076:enumerate
2740:Hence, if
380:, we have
184:set system
150:, and the
63:set theory
5906:Δ
5852:: 399–400
5809:202558957
5698:0024-6107
5645:0209-9683
5608:221818818
5562:201314765
5489:β
5463:ω
5439:⊆
5352:ω
5325:⊆
5277:β
5231:ω
5183:α
5180:∩
5175:α
5155:α
5123:ω
5113:α
5107:
5064:⟩
5055:ω
5048:α
5040:α
5032:⟨
5012:Δ
4986:ω
4959:ω
4948:countable
4928:ω
4884:that the
4850:Δ
4839:-system.
4827:Δ
4797:Δ
4785:-system.
4773:Δ
4753:Δ
4731:Δ
4626:
4568:≤
4484:∗
4426:over all
4353:∗
4301:∗
4254:∗
4201:−
4171:−
4147:≥
4115:Theorem.
4074:−
4009:∗
3977:∗
3964:≤
3942:−
3914:∗
3886:−
3841:∗
3813:−
3767:−
3741:−
3675:−
3640:−
3608:−
3597:a set of
3536:≤
3520:−
3506:Theorem.
3419:
3366:
3314:
3308:
3293:
3237:≤
3185:for some
3163:≤
3023:≤
2931:−
2913:−
2901:≤
2791:−
2773:−
2761:≥
2722:−
2707:−
2689:−
2677:≤
2661:−
2646:−
2628:−
2616:≤
2607:−
2592:−
2564:≤
2536:∈
2529:∑
2525:≤
2412:∈
2383:−
2368:−
2202:−
2093:∈
2080:∈
2074:∣
2062:∖
2017:∈
2006:for some
1979:. Define
1961:−
1949:≤
1846:∪
1843:⋯
1840:∪
1827:∪
1639:∩
1606:∈
1590:…
1461:−
1455:≤
1361:−
1246:−
1231:≤
862:For each
713:∈
576:, may be
391:∩
263:Δ
252:sunflower
210:of a set
162:Suppose
132:Δ
82:Δ
71:sunflower
5860:(2020),
5761:(1980),
5749:(2003),
5672:Rado, R.
5661:14043028
5615:Deza, M.
5505:See also
5408:′
5321:′
5309:. Build
1319:integers
632:integers
582:disjoint
319:distinct
5740:2238686
5706:0111692
5653:0637827
5600:4240687
5511:Cap set
5396:are in
5141:, let
5096:. For
5004:-sized
4892: (
4875:forcing
4706:of the
2815:, then
1024:, then
661:integer
536:of the
275:-system
208:subsets
94:-system
5992:
5807:
5771:
5738:
5728:
5704:
5696:
5659:
5651:
5643:
5606:
5598:
5560:
5550:
5202:, fix
5198:. By
4919:is an
4890:Shanin
4809:-lemma
4743:-lemma
4109:sets.
2032:to be
1426:sets.
144:-lemma
106:kernel
5886:Notes
5820:(PDF)
5805:S2CID
5657:S2CID
5604:S2CID
5578:arXiv
5558:S2CID
5530:arXiv
5421:then
4878:poset
2117:Then
850:Erdős
748:, if
578:empty
250:is a
186:over
182:is a
121:large
5990:ISBN
5791:2020
5769:ISBN
5726:ISBN
5694:ISSN
5641:ISSN
5548:ISBN
5381:<
5051:<
4894:1946
4842:The
4788:The
4241:and
3259:for
3196:>
3056:>
2982:>
2963:The
2191:has
1803:Let
1785:<
1297:and
899:>
873:>
854:Rado
852:and
534:some
340:and
254:(or
98:sets
69:, a
65:and
6121:doi
6028:doi
5982:doi
5832:doi
5795:doi
5718:doi
5686:doi
5633:doi
5588:doi
5574:344
5540:doi
5340:of
5289:on
5164:sup
4899:If
4896:).
4623:log
4545:is
4446:in
4315:to
3416:log
3363:log
3311:log
3305:log
3290:log
969:of
360:in
297:of
73:or
6167::
6117:12
6115:.
6111:.
6062:.
6024:12
6022:.
6018:.
5988:,
5976:,
5938:.
5927:^
5850:53
5828:60
5826:,
5822:,
5803:,
5789:,
5785:,
5736:MR
5734:,
5724:,
5702:MR
5700:,
5692:,
5682:35
5670:;
5655:,
5649:MR
5647:,
5639:,
5627:,
5617:;
5602:,
5596:MR
5594:,
5586:,
5572:,
5556:,
5546:,
5538:,
5104:cf
4554:10
3996:.
3503:.
3437:.
2997:,
2725:1.
2501::
2398:.
1800:.
1064:.
888:,
146:,
6154:.
6129:.
6123::
6097:.
6085:.
6072:.
6049:.
6036:.
6030::
5984::
5949:.
5834::
5797::
5720::
5688::
5635::
5629:1
5590::
5580::
5542::
5532::
5467:1
5442:j
5434:i
5430:A
5405:S
5384:j
5378:i
5356:2
5328:S
5318:S
5297:S
5257:f
5235:2
5210:S
5186:)
5171:A
5167:(
5161:=
5158:)
5152:(
5149:f
5127:1
5119:=
5116:)
5110:(
5084:W
5059:2
5045::
5036:A
4990:2
4963:2
4932:2
4907:W
4688:0
4684:C
4680:A
4661:3
4635:)
4632:n
4629:(
4619:n
4586:)
4583:3
4580:,
4577:k
4574:(
4571:f
4562:2
4559:k
4533:3
4530:=
4527:r
4503:|
4499:F
4495:|
4490:|
4480:F
4475:|
4454:F
4434:A
4412:A
4408:F
4385:A
4381:F
4359:|
4349:F
4344:|
4323:A
4297:F
4276:A
4250:F
4229:F
4207:)
4204:1
4198:)
4195:r
4192:,
4189:b
4186:(
4183:f
4180:(
4177:)
4174:1
4168:)
4165:r
4162:,
4159:a
4156:(
4153:f
4150:(
4144:)
4141:r
4138:,
4135:b
4132:+
4129:a
4126:(
4123:f
4097:r
4077:1
4071:r
4051:r
4031:r
4005:H
3983:|
3973:H
3968:|
3960:|
3956:H
3952:|
3948:)
3945:1
3939:r
3936:(
3910:H
3889:1
3883:r
3863:H
3837:H
3816:1
3810:r
3790:H
3770:1
3764:r
3744:1
3738:r
3718:H
3698:r
3678:1
3672:k
3652:)
3649:r
3646:,
3643:1
3637:k
3634:(
3631:h
3611:1
3605:r
3585:1
3582:=
3579:k
3557:.
3554:)
3551:r
3548:,
3545:k
3542:(
3539:f
3531:k
3527:)
3523:1
3517:r
3514:(
3491:r
3471:)
3468:r
3465:,
3462:k
3459:(
3456:f
3425:)
3422:k
3413:r
3410:(
3407:O
3404:=
3401:C
3381:)
3378:)
3375:k
3372:r
3369:(
3360:r
3357:(
3354:O
3351:=
3348:C
3338:.
3326:)
3323:)
3320:k
3317:(
3302:)
3299:k
3296:(
3285:3
3281:r
3277:(
3274:O
3271:=
3268:C
3245:k
3241:C
3234:)
3231:r
3228:,
3225:k
3222:(
3219:f
3199:0
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3160:)
3157:3
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3148:(
3145:f
3125:3
3122:=
3119:r
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3079:r
3059:0
3053:C
3031:k
3027:C
3020:)
3017:r
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2102:.
2099:}
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2077:a
2071:}
2068:a
2065:{
2059:S
2056:{
2053:=
2048:a
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2020:A
2014:a
1992:a
1988:W
1967:)
1964:1
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1955:(
1952:k
1946:t
1943:k
1923:A
1903:k
1881:i
1877:A
1854:t
1850:A
1835:2
1831:A
1822:1
1818:A
1814:=
1811:A
1788:r
1782:t
1762:r
1742:W
1720:i
1716:A
1695:W
1675:j
1672:=
1669:i
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1643:A
1634:i
1630:A
1609:W
1601:t
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1593:,
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1582:2
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1000:(
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618:)
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606:(
603:f
564:S
544:W
520:U
500:W
480:S
460:U
440:W
420:W
400:S
397:=
394:B
388:A
368:W
348:B
328:A
305:U
285:S
238:W
218:U
194:U
170:W
50:.
31::
20:)
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