746:
3317:
672:
3298:
3329:
Aigner and
Ziegler call the final one of their five proofs "the most beautiful of them all". Its origins are unclear, but the approach is often referred to as Zykov Symmetrization as it was used in Zykov's proof of a generalization of Turán's Theorem . This proof goes by taking a
1820:
1583:, which is a collection of independent sets, with edges between each two vertices from different independent sets. A simple calculation shows that the number of edges of this graph is maximized when all independent set sizes are as close to equal as possible.
2163:(and adjusting edges accordingly) would increase the value of the sum. This can be seen by examining the changes to either side of the above expression for the number of edges, or by noting that the degree of the moved vertex increases.
5275:
3169:
3730:
give that any maximal graph the same form as a Turán graph. As in the maximal degree vertex proof, a simple calculation shows that the number of edges is maximized when all independent set sizes are as close to equal as possible.
2359:
4393:
1682:
2616:
4185:
474:
1882:
626:
5044:
5760:
2867:
2081:
5915:
The lower bound was proven by
Razborov (2008) for the case of triangles, and was later generalized to all cliques by Reiher (2016). The upper bound is a consequence of the Kruskal–Katona theorem .
1938:
1989:
5467:
An issue with answering this question is that for a given density, there may be some bound not attained by any graph, but approached by some infinite sequence of graphs. To deal with this,
3052:
2418:
1675:
5851:
3831:
4064:
4016:
2973:
2907:
1151:
5408:
1677:
be the independent sets of the multipartite graph. Since two vertices have an edge between them if and only if they are not in the same independent set, the number of edges is
3272:
3061:
566:
5328:
4496:
2786:
2734:
1050:
4918:
5103:
2682:
2244:
5574:
2249:
3948:
2485:
5643:
4281:
4225:
1295:
4806:
4696:
4628:
4549:
4110:
3896:
3717:
3397:
3361:
3210:
2206:
2107:
1507:
844:
344:
140:
4980:
4868:
533:
379:
218:
5910:
5883:
5670:
5605:
5523:
5462:
5435:
5355:
5130:
4945:
4833:
4723:
3858:
3542:
2161:
2134:
2019:
1554:
958:
891:
107:
4422:
4306:
4749:
4575:
3763:
3620:
3594:
3568:
3427:
2511:
1004:
739:
5800:
5780:
5694:
5547:
5496:
5150:
5068:
4891:
4773:
4651:
4595:
4516:
4442:
4301:
4249:
3975:
3783:
3680:
3660:
3640:
3507:
3487:
3467:
3447:
3292:
3237:
2927:
2806:
2636:
2438:
1613:
1574:
1527:
1474:
1454:
1434:
1411:
1391:
1371:
1351:
1331:
1254:
1231:
1211:
1191:
1171:
1070:
978:
931:
911:
864:
807:
787:
767:
713:
693:
661:
498:
399:
311:
291:
180:
160:
71:
5155:
2516:
404:
5782:-partite graph where all parts but the unique smallest part have the same size, and sizes of the parts are chosen such that the total edge density is
182:
parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the
5046:. A paper by Alon and Shikhelman in 2016 gives the following generalization, which is similar to the Erdos-Stone generalization of Turán's theorem:
6035:
3954:: not only does it contain a triangle, it must also contain cycles of all other possible lengths up to the number of vertices in the graph.
4115:
3310:
4022:
1827:
1815:{\displaystyle \sum _{i\neq j}\left|S_{i}\right|\left|S_{j}\right|={\frac {1}{2}}\left(n^{2}-\sum _{i}\left|S_{i}\right|^{2}\right),}
571:
6405:
5437:
s. One could ask the far more general question: if you are given the edge density of a graph, what can you say about the density of
4985:
5699:
2819:
2024:
1824:
where the left hand side follows from direct counting, and the right hand side follows from complementary counting. To show the
6096:
6400:
3055:
639:
list five different proofs of Turán's theorem. Many of the proofs involve reducing to the case where the graph is a complete
2813:
1885:
6196:
5975:
Mantel, W. (1907), "Problem 28 (Solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff)",
1891:
1943:
5924:
4188:
4654:
47:
1591:
This proof, as well as the Zykov
Symmetrization proof, involve reducing to the case where the graph is a complete
3899:
2998:
2364:
1621:
5809:
3792:
4028:
3980:
6192:
2932:
2872:
1078:
5372:
4018:
3363:-free graph, and applying steps to make it more similar to the Turán Graph while increasing edge count.
3309:
and bounding the size of the chosen set using the Cauchy–Schwarz inequality proves Turán's theorem. See
74:
43:
39:
3242:
745:
538:
46:, an area studying the largest or smallest graphs with given properties, and is a special case of the
6017:
6013:
5366:
5283:
4451:
3723:
2739:
2687:
2441:
1016:
20:
5475:
are often considered. In particular, graphons contain the limit of any infinite sequence of graphs.
4896:
5073:
3786:
2641:
2211:
258:
5552:
6363:
6345:
6318:
6277:
6257:
6231:
6123:
3904:
3176:
2450:
1592:
640:
4873:
This was first shown by Zykov (1949) using Zykov
Symmetrization. Since the Turán Graph contains
6295:
6159:
6031:
5610:
4257:
4194:
3727:
1259:
477:
4778:
4668:
4600:
4521:
4082:
3866:
3689:
3369:
3333:
3182:
2178:
2086:
1479:
816:
316:
112:
6355:
6310:
6267:
6215:
6171:
6105:
6087:
6023:
4950:
4838:
4252:
3306:
503:
349:
188:
35:
6227:
6119:
6074:
5888:
5856:
5648:
5583:
5501:
5440:
5413:
5333:
5108:
4923:
4811:
4701:
3836:
3515:
2139:
2112:
1997:
1532:
936:
869:
80:
6223:
6115:
6091:
6070:
4665:
Another natural extension of Turán's theorem is the following question: if a graph has no
4398:
3951:
2983:
The key claim in this proof was independently found by Caro and Wei. This proof is due to
1615:
independent sets of size as close as possible to equal. This step can be done as follows:
5950:
5270:{\displaystyle (1+o(1)){\binom {\chi (H)-1}{a}}\left({\frac {n}{\chi (H)-1}}\right)^{a}.}
4728:
4633:
The general question of how many edges can be included in a graph without a copy of some
4554:
3742:
3599:
3573:
3547:
3406:
3164:{\displaystyle S={\frac {1}{d_{1}+1}}+{\frac {1}{d_{2}+1}}+\cdots +{\frac {1}{d_{n}+1}}.}
2490:
983:
718:
250:
3863:
A strengthened form of Mantel's theorem states that any
Hamiltonian graph with at least
3316:
3297:
671:
220:. Turán's theorem states that the Turán graph has the largest number of edges among all
5785:
5765:
5679:
5532:
5481:
5468:
5135:
5053:
4876:
4758:
4636:
4580:
4501:
4427:
4286:
4234:
3960:
3768:
3665:
3645:
3625:
3492:
3472:
3452:
3432:
3277:
3222:
2912:
2791:
2621:
2423:
1598:
1580:
1559:
1512:
1459:
1439:
1419:
1396:
1376:
1356:
1336:
1316:
1239:
1216:
1196:
1176:
1156:
1055:
963:
916:
896:
849:
792:
772:
752:
698:
678:
646:
483:
384:
296:
276:
240:
183:
165:
145:
56:
5927:, a generalization of Turán's theorem from forbidden cliques to forbidden Turán graphs
1476:. This increases the number of edges by our maximality assumption and keeps the graph
6394:
6188:
6175:
6127:
6050:
6009:
2354:{\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=\sum _{i,j\ {\text{adjacent}}}x_{i}x_{j}}
1310:
247:
6367:
6322:
5576:
of the vertices, and connect two vertices if and only if they are in the chosen set.
6281:
2988:
2788:
without decreasing the value of the function. Hence, there is a point with at most
254:
27:
6235:
6027:
6359:
6272:
50:
on the maximum number of edges in a graph that does not have a given subgraph.
6314:
6054:
4388:{\displaystyle \left(1-{\frac {1}{\chi (H)-1}}+o(1)\right){\frac {n^{2}}{2}}}
2984:
6219:
6200:
6110:
1579:
Repeating this argument eventually produces a graph in the same form as a
4025:(the minimum number of cliques needed to cover all its edges) is at most
3957:
Another strengthening of Mantel's theorem states that the edges of every
6248:
Alon, Noga; Shikhelman, Clara (2016), "Many T copies in H-free graphs",
5472:
3215:
Select every vertex that is adjacent to none of the vertices before it.
2021:
differ by more than one in size. In particular, supposing that we have
244:
2611:{\displaystyle f(x_{1},\ldots ,x_{i}-t,\ldots ,x_{j}+t,\ldots ,x_{n})}
893:(which exists by maximality), and partition the vertices into the set
4180:{\displaystyle \left(1-{\frac {1}{r}}+o(1)\right){\frac {n^{2}}{2}}}
469:{\displaystyle \left(1-{\frac {1}{r}}+o(1)\right){\frac {n^{2}}{2}}}
6094:(1965), "Maxima for graphs and a new proof of a theorem of Turán",
3509:. Repeat this until all non-adjacent vertices have the same degree.
1595:, and showing that the number of edges is maximized when there are
643:, and showing that the number of edges is maximized when there are
6350:
6262:
4021:
which are either edges or triangles. As a corollary, the graph's
265:; it was stated in 1907 by Willem Mantel, a Dutch mathematician.
3171:
The proof attempts to find such an independent set as follows:
1994:
To prove the Turán Graph is optimal, one can argue that no two
1877:{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}}
621:{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}}
5039:{\displaystyle {\binom {r}{a}}\left({\frac {n}{r}}\right)^{a}}
243:
giving its extreme case, were first described and studied by
6140:
Zykov, A. (1949), "On some properties of linear complexes",
5755:{\displaystyle 1-{\frac {1}{t-1}}<d\leq 1-{\frac {1}{t}}}
4079:
Turán's theorem shows that the largest number of edges in a
3833:
In other words, one must delete nearly half of the edges in
568:. Many of the following proofs only give the upper bound of
2862:{\displaystyle {\frac {1}{2}}\left(1-{\frac {1}{r}}\right)}
2076:{\displaystyle \left|S_{i}\right|\geq \left|S_{j}\right|+2}
346:
as a subgraph has at most as many edges as the Turán graph
3311:
Method of conditional probabilities § Turán's theorem
6336:
Reiher, Christian (2016), "The clique density theorem",
4283:. The largest possible number of edges in a graph where
4518:, so the Turán graph establishes the lower bound. As a
3765:
is Mantel's theorem: The maximum number of edges in an
1946:
1894:
34:
bounds the number of edges that can be included in an
16:
Extremal graph theory bound on clique-free graph edges
5891:
5859:
5812:
5788:
5768:
5702:
5682:
5651:
5613:
5586:
5555:
5535:
5504:
5484:
5443:
5416:
5375:
5336:
5286:
5158:
5138:
5111:
5076:
5056:
4988:
4953:
4926:
4899:
4879:
4841:
4814:
4781:
4761:
4731:
4704:
4671:
4639:
4603:
4583:
4557:
4524:
4504:
4454:
4430:
4401:
4309:
4289:
4260:
4237:
4197:
4118:
4085:
4031:
3983:
3963:
3907:
3869:
3839:
3795:
3771:
3745:
3692:
3668:
3648:
3628:
3602:
3576:
3550:
3518:
3495:
3475:
3455:
3435:
3409:
3372:
3336:
3280:
3245:
3225:
3185:
3064:
3001:
2935:
2915:
2875:
2822:
2794:
2742:
2690:
2644:
2624:
2519:
2493:
2453:
2426:
2367:
2252:
2214:
2181:
2142:
2115:
2089:
2027:
2000:
1830:
1685:
1624:
1601:
1562:
1535:
1515:
1482:
1462:
1442:
1422:
1399:
1379:
1359:
1339:
1319:
1262:
1242:
1219:
1199:
1179:
1159:
1081:
1058:
1019:
986:
966:
939:
919:
899:
872:
852:
819:
795:
775:
755:
721:
701:
681:
649:
574:
541:
506:
486:
407:
387:
352:
319:
299:
279:
191:
168:
148:
115:
83:
59:
42:
of a given size. It is one of the central results of
6201:"The representation of a graph by set intersections"
5676:
Take a number of vertices approaching infinity. Let
1933:{\textstyle \sum \limits _{i}\left|S_{i}\right|^{2}}
6022:(6th ed.), Springer-Verlag, pp. 285–289,
2808:nonzero variables where the function is maximized.
5953:(1941), "On an extremal problem in graph theory",
5904:
5877:
5845:
5794:
5774:
5754:
5688:
5664:
5637:
5599:
5568:
5541:
5517:
5490:
5456:
5429:
5402:
5349:
5322:
5269:
5144:
5124:
5097:
5062:
5038:
4974:
4939:
4912:
4885:
4862:
4827:
4800:
4767:
4743:
4717:
4690:
4645:
4622:
4589:
4569:
4543:
4510:
4490:
4436:
4416:
4387:
4295:
4275:
4243:
4219:
4179:
4104:
4058:
4010:
3969:
3942:
3890:
3852:
3825:
3777:
3757:
3711:
3674:
3654:
3634:
3614:
3588:
3562:
3536:
3501:
3481:
3461:
3441:
3421:
3391:
3355:
3286:
3266:
3231:
3204:
3163:
3046:
2967:
2921:
2901:
2861:
2800:
2780:
2728:
2676:
2630:
2610:
2505:
2479:
2432:
2412:
2353:
2238:
2200:
2155:
2128:
2101:
2075:
2013:
1984:{\textstyle \sum \limits _{i}\left|S_{i}\right|=n}
1983:
1932:
1876:
1814:
1669:
1607:
1568:
1548:
1521:
1501:
1468:
1448:
1428:
1405:
1385:
1365:
1345:
1325:
1289:
1248:
1225:
1205:
1185:
1165:
1145:
1064:
1044:
998:
972:
952:
925:
905:
885:
866:vertices with the maximal number of edges. Find a
858:
838:
801:
781:
761:
733:
707:
687:
655:
620:
560:
527:
492:
468:
393:
373:
338:
305:
285:
212:
174:
154:
134:
101:
65:
5216:
5186:
5005:
4992:
1036:
1023:
5365:Turan's theorem states that if a graph has edge
2995:. The proof shows that every graph with degrees
6296:"On the minimal density of triangles in graphs"
4725:can it have? Turán's theorem is the case where
4577:, Turán's theorem is the special case in which
1556:-free, so the same argument can be repeated on
500:gets larger, the fraction of edges included in
3399:-free graph, the following steps are applied:
3320:(Zykov Symmetrization) Example of second step.
749:(Maximal Degree Vertex) Deleting edges within
6016:(2018), "Chapter 41: Turán's graph theorem",
3301:(Zykov Symmetrization) Example of first step.
2172:
663:parts of size as close as possible to equal.
636:
8:
4053:
4032:
4005:
3984:
3817:
3796:
1940:term on the right hand side suffices, since
3719:free while increasing the number of edges.
2513:are not adjacent in the graph, the function
1009:Now, one can bound edges above as follows:
6349:
6271:
6261:
6250:Journal of Combinatorial Theory, Series B
6164:Journal of Combinatorial Theory, Series B
6109:
5896:
5890:
5858:
5825:
5811:
5787:
5767:
5742:
5709:
5701:
5681:
5656:
5650:
5622:
5618:
5612:
5591:
5585:
5556:
5554:
5534:
5509:
5503:
5483:
5448:
5442:
5421:
5415:
5382:
5374:
5341:
5335:
5285:
5258:
5227:
5215:
5185:
5183:
5157:
5137:
5116:
5110:
5075:
5055:
5030:
5016:
5004:
4991:
4989:
4987:
4952:
4931:
4925:
4900:
4898:
4878:
4840:
4819:
4813:
4786:
4780:
4760:
4730:
4709:
4703:
4676:
4670:
4638:
4608:
4602:
4582:
4556:
4529:
4523:
4503:
4453:
4429:
4400:
4374:
4368:
4321:
4308:
4288:
4259:
4236:
4208:
4196:
4166:
4160:
4130:
4117:
4090:
4084:
4045:
4039:
4030:
3997:
3991:
3982:
3962:
3930:
3916:
3912:
3906:
3880:
3874:
3868:
3844:
3838:
3809:
3803:
3794:
3770:
3744:
3697:
3691:
3667:
3647:
3627:
3601:
3575:
3549:
3517:
3494:
3474:
3454:
3434:
3408:
3377:
3371:
3341:
3335:
3279:
3246:
3244:
3224:
3190:
3184:
3143:
3133:
3109:
3099:
3081:
3071:
3063:
3047:{\displaystyle d_{1},d_{2},\ldots ,d_{n}}
3038:
3019:
3006:
3000:
2957:
2947:
2939:
2936:
2934:
2929:gives that the maximal value is at least
2914:
2889:
2880:
2874:
2844:
2823:
2821:
2816:gives that the maximal value is at most
2793:
2769:
2756:
2741:
2711:
2698:
2689:
2665:
2652:
2643:
2623:
2599:
2574:
2549:
2530:
2518:
2492:
2471:
2458:
2452:
2425:
2413:{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
2404:
2385:
2372:
2366:
2345:
2335:
2324:
2311:
2295:
2276:
2263:
2251:
2246:, and considering maximizing the function
2213:
2186:
2180:
2147:
2141:
2120:
2114:
2088:
2057:
2036:
2026:
2005:
1999:
1965:
1951:
1945:
1924:
1914:
1899:
1893:
1863:
1857:
1842:
1829:
1798:
1788:
1773:
1760:
1741:
1728:
1710:
1690:
1684:
1670:{\displaystyle S_{1},S_{2},\ldots ,S_{r}}
1661:
1642:
1629:
1623:
1600:
1561:
1540:
1534:
1514:
1487:
1481:
1461:
1441:
1421:
1398:
1378:
1358:
1338:
1318:
1261:
1241:
1218:
1198:
1178:
1158:
1105:
1097:
1080:
1057:
1035:
1022:
1020:
1018:
985:
965:
944:
938:
918:
898:
877:
871:
851:
824:
818:
794:
774:
754:
720:
700:
680:
648:
607:
601:
586:
573:
548:
540:
505:
485:
455:
449:
419:
406:
386:
351:
324:
318:
298:
278:
190:
167:
147:
142:may be formed by partitioning the set of
120:
114:
82:
58:
6303:Combinatorics, Probability and Computing
6057:[On the graph theorem of Turán]
5970:
5968:
5846:{\displaystyle d\leq 1-{\frac {1}{r-1}}}
5330:attains the desired number of copies of
4751:. Zykov's Theorem answers this question:
3977:-vertex graph may be covered by at most
3826:{\displaystyle \lfloor n^{2}/4\rfloor .}
3739:The special case of Turán's theorem for
3315:
3296:
813:This was Turán's original proof. Take a
744:
670:
273:Turán's theorem states that every graph
5945:
5943:
5941:
5937:
4059:{\displaystyle \lfloor n^{2}/4\rfloor }
4011:{\displaystyle \lfloor n^{2}/4\rfloor }
2447:The idea behind their proof is that if
3274:, so this process gives an average of
1301:Adding these bounds gives the result.
4808:s and the largest possible number of
3239:is included in this with probability
7:
6004:
6002:
6000:
5998:
5996:
5994:
5992:
5990:
5549:approaching infinity. Pick a set of
2968:{\displaystyle {\frac {|E|}{n^{2}}}}
2902:{\displaystyle x_{i}={\frac {1}{n}}}
1333:of largest degree. Consider the set
675:(Induction on n) An example of sets
5498:, the construction for the largest
5280:As in Erdős–Stone, the Turán graph
1948:
1896:
1146:{\displaystyle (r-1)|B|=(r-1)(n-r)}
5645:The construction for the smallest
5403:{\displaystyle 1-{\frac {1}{r-1}}}
5190:
4996:
4191:finds the number of edges up to a
3686:All of these steps keep the graph
2208:free graph with vertices labelled
1587:Complete Multipartite Optimization
1256:is at most the number of edges of
1027:
480:. Intuitively, this means that as
14:
5105:. The largest possible number of
5070:be a graph with chromatic number
4448:One can see that the Turán graph
3860:to obtain a triangle-free graph.
4303:does not appear as a subgraph is
3267:{\displaystyle {\frac {1}{d+1}}}
2638:. Hence, one can either replace
2440:. This function is known as the
561:{\displaystyle 1-{\frac {1}{r}}}
77:graph that does not contain any
6382:Large networks and graph limits
6208:Canadian Journal of Mathematics
6097:Canadian Journal of Mathematics
5323:{\displaystyle T(n,\chi (H)-1)}
4755:(Zykov's Theorem) The graph on
4491:{\displaystyle T(n,\chi (H)-1)}
2781:{\displaystyle (0,x_{i}+x_{j})}
2729:{\displaystyle (x_{i}+x_{j},0)}
1045:{\displaystyle {\binom {r}{2}}}
313:vertices that does not contain
6162:(1971), "Pancyclic graphs I",
5872:
5860:
5317:
5308:
5302:
5290:
5242:
5236:
5201:
5195:
5180:
5177:
5171:
5159:
5086:
5080:
4969:
4957:
4913:{\displaystyle {\frac {n}{r}}}
4857:
4845:
4485:
4476:
4470:
4458:
4411:
4405:
4360:
4354:
4336:
4330:
4270:
4264:
4214:
4201:
4152:
4146:
3429:are non-adjacent vertices and
2948:
2940:
2775:
2743:
2723:
2691:
2671:
2645:
2605:
2523:
2301:
2256:
2175:. They begin by considering a
1284:
1266:
1140:
1128:
1125:
1113:
1106:
1098:
1094:
1082:
522:
510:
441:
435:
368:
356:
207:
195:
96:
84:
1:
6325:– via MathSciNet (AMS).
5853:, this gives a graph that is
5410:, it has a nonzero number of
5132:s in a graph with no copy of
5098:{\displaystyle \chi (H)>a}
4498:cannot contain any copies of
2677:{\displaystyle (x_{i},x_{j})}
2239:{\displaystyle 1,2,\ldots ,n}
1416:Now, delete all edges within
6294:Razborov, Alexander (2008).
6176:10.1016/0095-8956(71)90016-5
6028:10.1007/978-3-662-57265-8_41
5955:Matematikai és Fizikai Lapok
5885:-partite and hence gives no
5569:{\displaystyle {\sqrt {d}}N}
5050:(Alon-Shikhelman, 2016) Let
3722:Now, non-adjacency forms an
3622:adjacent, then replace both
3294:vertices in the chosen set.
2975:, giving the desired bound.
2444:of the graph and its edges.
1353:of vertices not adjacent to
1297:by the inductive hypothesis.
6360:10.4007/annals.2016.184.3.1
6144:, New Series (in Russian),
4661:Maximizing Other Quantities
3943:{\displaystyle K_{n/2,n/2}}
2480:{\displaystyle x_{i},x_{j}}
2173:Motzkin & Straus (1965)
1436:and draw all edges between
1236:The number of edges within
637:Aigner & Ziegler (2018)
6422:
6273:10.1016/j.jctb.2016.03.004
6055:"Turán Pál gráf tételéről"
5529:Take a number of vertices
4655:forbidden subgraph problem
4227:error in all other graphs:
3305:Applying this fact to the
769:and drawing edges between
535:gets closer and closer to
48:forbidden subgraph problem
23:in analytic number theory.
18:
6315:10.1017/S0963548308009085
5696:be the integer such that
5478:For a given edge density
4424:constant only depends on
4075:Other Forbidden Subgraphs
3898:edges must either be the
3449:has a higher degree than
2814:Cauchy–Schwarz inequality
2109:, moving one vertex from
1886:Cauchy–Schwarz inequality
239:Turán's theorem, and the
6406:Theorems in graph theory
5638:{\displaystyle d^{k/2}.}
4276:{\displaystyle \chi (H)}
4220:{\displaystyle o(n^{2})}
3900:complete bipartite graph
2993:The Probabilistic Method
1393:of vertices adjacent to
1290:{\displaystyle T(n-r,r)}
19:Not to be confused with
4893:parts with size around
4801:{\displaystyle K_{r+1}}
4691:{\displaystyle K_{r+1}}
4623:{\displaystyle K_{r+1}}
4544:{\displaystyle K_{r+1}}
4105:{\displaystyle K_{r+1}}
3891:{\displaystyle n^{2}/4}
3734:
3712:{\displaystyle K_{r+1}}
3392:{\displaystyle K_{r+1}}
3366:In particular, given a
3356:{\displaystyle K_{r+1}}
3205:{\displaystyle K_{r+1}}
2487:are both nonzero while
2201:{\displaystyle K_{r+1}}
2102:{\displaystyle i\neq j}
1502:{\displaystyle K_{r+1}}
839:{\displaystyle K_{r+1}}
381:. For a fixed value of
339:{\displaystyle K_{r+1}}
135:{\displaystyle K_{r+1}}
6220:10.4153/CJM-1966-014-3
6111:10.4153/CJM-1965-053-6
5906:
5879:
5847:
5804:
5796:
5776:
5756:
5690:
5672:density is as follows:
5666:
5639:
5601:
5578:
5570:
5543:
5525:density is as follows:
5519:
5492:
5458:
5431:
5404:
5351:
5324:
5278:
5271:
5146:
5126:
5099:
5064:
5040:
4976:
4975:{\displaystyle T(n,r)}
4941:
4914:
4887:
4871:
4864:
4863:{\displaystyle T(n,r)}
4829:
4802:
4769:
4745:
4719:
4698:s, how many copies of
4692:
4647:
4624:
4591:
4571:
4545:
4512:
4492:
4446:
4438:
4418:
4389:
4297:
4277:
4245:
4231:(Erdős–Stone) Suppose
4221:
4181:
4106:
4060:
4012:
3971:
3944:
3892:
3854:
3827:
3779:
3759:
3713:
3676:
3656:
3636:
3616:
3590:
3564:
3538:
3503:
3483:
3463:
3443:
3423:
3393:
3357:
3321:
3302:
3288:
3268:
3233:
3206:
3165:
3048:
2969:
2923:
2903:
2863:
2802:
2782:
2730:
2678:
2632:
2612:
2507:
2481:
2434:
2414:
2355:
2240:
2202:
2157:
2130:
2103:
2077:
2015:
1985:
1934:
1878:
1816:
1671:
1609:
1570:
1550:
1523:
1503:
1470:
1450:
1430:
1407:
1387:
1367:
1347:
1327:
1291:
1250:
1227:
1213:can connect to all of
1207:
1187:
1167:
1147:
1066:
1046:
1000:
974:
954:
927:
907:
887:
860:
840:
810:
803:
783:
763:
742:
735:
709:
689:
657:
622:
562:
529:
528:{\displaystyle T(n,r)}
494:
470:
395:
375:
374:{\displaystyle T(n,r)}
340:
307:
287:
214:
213:{\displaystyle T(n,r)}
176:
156:
136:
103:
67:
6401:Extremal graph theory
6338:Annals of Mathematics
5907:
5905:{\displaystyle K_{r}}
5880:
5878:{\displaystyle (r-1)}
5848:
5797:
5777:
5757:
5691:
5674:
5667:
5665:{\displaystyle K_{r}}
5640:
5602:
5600:{\displaystyle K_{r}}
5571:
5544:
5527:
5520:
5518:{\displaystyle K_{r}}
5493:
5459:
5457:{\displaystyle K_{r}}
5432:
5430:{\displaystyle K_{r}}
5405:
5352:
5350:{\displaystyle K_{a}}
5325:
5272:
5147:
5127:
5125:{\displaystyle K_{a}}
5100:
5065:
5048:
5041:
4977:
4942:
4940:{\displaystyle K_{a}}
4915:
4888:
4865:
4835:s is the Turán graph
4830:
4828:{\displaystyle K_{a}}
4803:
4770:
4753:
4746:
4720:
4718:{\displaystyle K_{a}}
4693:
4648:
4625:
4592:
4572:
4551:has chromatic number
4546:
4513:
4493:
4439:
4419:
4390:
4298:
4278:
4246:
4229:
4222:
4182:
4107:
4061:
4013:
3972:
3945:
3893:
3855:
3853:{\displaystyle K_{n}}
3828:
3780:
3760:
3714:
3677:
3657:
3637:
3617:
3591:
3565:
3539:
3537:{\displaystyle u,v,w}
3504:
3484:
3464:
3444:
3424:
3394:
3358:
3319:
3300:
3289:
3269:
3234:
3207:
3179:of the vertices of a
3166:
3049:
2970:
2924:
2904:
2864:
2803:
2783:
2731:
2679:
2633:
2613:
2508:
2482:
2435:
2415:
2361:over all nonnegative
2356:
2241:
2203:
2171:This proof is due to
2158:
2156:{\displaystyle S_{i}}
2131:
2129:{\displaystyle S_{j}}
2104:
2078:
2016:
2014:{\displaystyle S_{i}}
1986:
1935:
1879:
1817:
1672:
1610:
1571:
1551:
1549:{\displaystyle K_{r}}
1524:
1504:
1471:
1451:
1431:
1408:
1388:
1368:
1348:
1328:
1309:This proof is due to
1305:Maximal Degree Vertex
1292:
1251:
1228:
1208:
1193:, since no vertex in
1188:
1168:
1148:
1067:
1047:
1001:
975:
955:
953:{\displaystyle K_{r}}
928:
908:
888:
886:{\displaystyle K_{r}}
861:
841:
804:
784:
764:
748:
736:
710:
690:
674:
658:
623:
563:
530:
495:
471:
396:
376:
341:
308:
288:
215:
177:
157:
137:
104:
102:{\displaystyle (r+1)}
68:
44:extremal graph theory
38:that does not have a
6019:Proofs from THE BOOK
5889:
5857:
5810:
5786:
5766:
5700:
5680:
5649:
5611:
5584:
5553:
5533:
5502:
5482:
5441:
5414:
5373:
5367:homomorphism density
5334:
5284:
5156:
5136:
5109:
5074:
5054:
4986:
4951:
4924:
4897:
4877:
4839:
4812:
4779:
4759:
4729:
4702:
4669:
4637:
4601:
4581:
4555:
4522:
4502:
4452:
4428:
4417:{\displaystyle o(1)}
4399:
4307:
4287:
4258:
4235:
4195:
4116:
4083:
4029:
3981:
3961:
3905:
3867:
3837:
3793:
3769:
3743:
3724:equivalence relation
3690:
3666:
3646:
3626:
3600:
3574:
3548:
3516:
3493:
3473:
3453:
3433:
3407:
3370:
3334:
3325:Zykov Symmetrization
3278:
3243:
3223:
3183:
3062:
2999:
2979:Probabilistic Method
2933:
2913:
2873:
2820:
2792:
2740:
2688:
2642:
2622:
2517:
2491:
2451:
2424:
2365:
2250:
2212:
2179:
2140:
2113:
2087:
2025:
1998:
1944:
1892:
1884:bound, applying the
1828:
1683:
1622:
1599:
1560:
1533:
1513:
1480:
1460:
1440:
1420:
1397:
1377:
1357:
1337:
1317:
1260:
1240:
1217:
1197:
1177:
1157:
1079:
1056:
1017:
984:
964:
937:
917:
897:
870:
850:
817:
793:
773:
753:
719:
699:
679:
647:
572:
539:
504:
484:
405:
385:
350:
317:
297:
277:
259:triangle-free graphs
189:
166:
146:
113:
81:
57:
5925:Erdős–Stone theorem
4744:{\displaystyle a=2}
4570:{\displaystyle r+1}
4189:Erdős–Stone theorem
4023:intersection number
3787:triangle-free graph
3758:{\displaystyle r=2}
3728:equivalence classes
3615:{\displaystyle u,w}
3589:{\displaystyle v,w}
3563:{\displaystyle u,v}
3422:{\displaystyle u,v}
3219:A vertex of degree
2506:{\displaystyle i,j}
999:{\displaystyle n-r}
734:{\displaystyle r=3}
257:of the theorem for
6014:Ziegler, Günter M.
5977:Wiskundige Opgaven
5902:
5875:
5843:
5792:
5772:
5752:
5686:
5662:
5635:
5597:
5566:
5539:
5515:
5488:
5454:
5427:
5400:
5361:Edge-Clique region
5347:
5320:
5267:
5142:
5122:
5095:
5060:
5036:
4972:
4937:
4910:
4883:
4860:
4825:
4798:
4765:
4741:
4715:
4688:
4643:
4620:
4587:
4567:
4541:
4508:
4488:
4434:
4414:
4385:
4293:
4273:
4241:
4217:
4177:
4102:
4056:
4008:
3967:
3940:
3888:
3850:
3823:
3775:
3755:
3709:
3672:
3652:
3632:
3612:
3586:
3560:
3544:are vertices with
3534:
3499:
3479:
3459:
3439:
3419:
3389:
3353:
3322:
3303:
3284:
3264:
3229:
3202:
3177:random permutation
3161:
3044:
2991:, from their book
2965:
2919:
2899:
2859:
2798:
2778:
2726:
2674:
2628:
2608:
2503:
2477:
2430:
2410:
2351:
2330:
2236:
2198:
2153:
2126:
2099:
2073:
2011:
1981:
1956:
1930:
1904:
1874:
1812:
1778:
1701:
1667:
1605:
1593:multipartite graph
1566:
1546:
1519:
1499:
1466:
1446:
1426:
1403:
1383:
1363:
1343:
1323:
1313:. Take the vertex
1287:
1246:
1223:
1203:
1183:
1163:
1143:
1075:There are at most
1062:
1042:
1013:There are exactly
996:
970:
950:
923:
903:
883:
856:
836:
811:
799:
779:
759:
743:
731:
705:
685:
653:
641:multipartite graph
618:
558:
525:
490:
466:
391:
371:
336:
303:
283:
210:
172:
152:
132:
99:
63:
6063:Matematikai Lapok
6037:978-3-662-57265-8
5841:
5795:{\displaystyle d}
5775:{\displaystyle t}
5750:
5725:
5689:{\displaystyle t}
5561:
5542:{\displaystyle N}
5491:{\displaystyle d}
5398:
5252:
5214:
5145:{\displaystyle H}
5063:{\displaystyle H}
5024:
5003:
4908:
4886:{\displaystyle r}
4775:vertices with no
4768:{\displaystyle n}
4646:{\displaystyle H}
4590:{\displaystyle H}
4511:{\displaystyle H}
4437:{\displaystyle H}
4383:
4346:
4296:{\displaystyle H}
4244:{\displaystyle H}
4175:
4138:
3970:{\displaystyle n}
3778:{\displaystyle n}
3675:{\displaystyle v}
3655:{\displaystyle w}
3635:{\displaystyle u}
3596:non-adjacent but
3502:{\displaystyle u}
3482:{\displaystyle v}
3462:{\displaystyle v}
3442:{\displaystyle u}
3287:{\displaystyle S}
3262:
3232:{\displaystyle d}
3156:
3122:
3094:
2963:
2922:{\displaystyle i}
2897:
2852:
2831:
2801:{\displaystyle r}
2631:{\displaystyle t}
2433:{\displaystyle 1}
2327:
2323:
2307:
1947:
1895:
1872:
1850:
1769:
1749:
1686:
1608:{\displaystyle r}
1569:{\displaystyle B}
1522:{\displaystyle B}
1469:{\displaystyle B}
1449:{\displaystyle A}
1429:{\displaystyle A}
1406:{\displaystyle v}
1386:{\displaystyle B}
1366:{\displaystyle v}
1346:{\displaystyle A}
1326:{\displaystyle v}
1249:{\displaystyle B}
1226:{\displaystyle A}
1206:{\displaystyle B}
1186:{\displaystyle B}
1166:{\displaystyle A}
1065:{\displaystyle A}
1034:
973:{\displaystyle B}
926:{\displaystyle r}
906:{\displaystyle A}
859:{\displaystyle n}
802:{\displaystyle B}
782:{\displaystyle A}
762:{\displaystyle A}
708:{\displaystyle B}
688:{\displaystyle A}
656:{\displaystyle r}
616:
594:
556:
493:{\displaystyle n}
478:little-o notation
464:
427:
394:{\displaystyle r}
306:{\displaystyle n}
286:{\displaystyle G}
175:{\displaystyle r}
155:{\displaystyle n}
66:{\displaystyle n}
53:An example of an
40:complete subgraph
6413:
6385:
6384:
6380:Lovász, László,
6377:
6371:
6370:
6353:
6333:
6327:
6326:
6300:
6291:
6285:
6284:
6275:
6265:
6245:
6239:
6238:
6205:
6185:
6179:
6178:
6156:
6150:
6149:
6137:
6131:
6130:
6113:
6084:
6078:
6077:
6065:(in Hungarian),
6060:
6047:
6041:
6040:
6006:
5985:
5984:
5972:
5963:
5962:
5957:(in Hungarian),
5947:
5911:
5909:
5908:
5903:
5901:
5900:
5884:
5882:
5881:
5876:
5852:
5850:
5849:
5844:
5842:
5840:
5826:
5801:
5799:
5798:
5793:
5781:
5779:
5778:
5773:
5761:
5759:
5758:
5753:
5751:
5743:
5726:
5724:
5710:
5695:
5693:
5692:
5687:
5671:
5669:
5668:
5663:
5661:
5660:
5644:
5642:
5641:
5636:
5631:
5630:
5626:
5606:
5604:
5603:
5598:
5596:
5595:
5575:
5573:
5572:
5567:
5562:
5557:
5548:
5546:
5545:
5540:
5524:
5522:
5521:
5516:
5514:
5513:
5497:
5495:
5494:
5489:
5463:
5461:
5460:
5455:
5453:
5452:
5436:
5434:
5433:
5428:
5426:
5425:
5409:
5407:
5406:
5401:
5399:
5397:
5383:
5356:
5354:
5353:
5348:
5346:
5345:
5329:
5327:
5326:
5321:
5276:
5274:
5273:
5268:
5263:
5262:
5257:
5253:
5251:
5228:
5221:
5220:
5219:
5210:
5189:
5151:
5149:
5148:
5143:
5131:
5129:
5128:
5123:
5121:
5120:
5104:
5102:
5101:
5096:
5069:
5067:
5066:
5061:
5045:
5043:
5042:
5037:
5035:
5034:
5029:
5025:
5017:
5010:
5009:
5008:
4995:
4981:
4979:
4978:
4973:
4946:
4944:
4943:
4938:
4936:
4935:
4920:, the number of
4919:
4917:
4916:
4911:
4909:
4901:
4892:
4890:
4889:
4884:
4869:
4867:
4866:
4861:
4834:
4832:
4831:
4826:
4824:
4823:
4807:
4805:
4804:
4799:
4797:
4796:
4774:
4772:
4771:
4766:
4750:
4748:
4747:
4742:
4724:
4722:
4721:
4716:
4714:
4713:
4697:
4695:
4694:
4689:
4687:
4686:
4652:
4650:
4649:
4644:
4629:
4627:
4626:
4621:
4619:
4618:
4596:
4594:
4593:
4588:
4576:
4574:
4573:
4568:
4550:
4548:
4547:
4542:
4540:
4539:
4517:
4515:
4514:
4509:
4497:
4495:
4494:
4489:
4443:
4441:
4440:
4435:
4423:
4421:
4420:
4415:
4394:
4392:
4391:
4386:
4384:
4379:
4378:
4369:
4367:
4363:
4347:
4345:
4322:
4302:
4300:
4299:
4294:
4282:
4280:
4279:
4274:
4253:chromatic number
4251:is a graph with
4250:
4248:
4247:
4242:
4226:
4224:
4223:
4218:
4213:
4212:
4186:
4184:
4183:
4178:
4176:
4171:
4170:
4161:
4159:
4155:
4139:
4131:
4111:
4109:
4108:
4103:
4101:
4100:
4065:
4063:
4062:
4057:
4049:
4044:
4043:
4017:
4015:
4014:
4009:
4001:
3996:
3995:
3976:
3974:
3973:
3968:
3949:
3947:
3946:
3941:
3939:
3938:
3934:
3920:
3897:
3895:
3894:
3889:
3884:
3879:
3878:
3859:
3857:
3856:
3851:
3849:
3848:
3832:
3830:
3829:
3824:
3813:
3808:
3807:
3784:
3782:
3781:
3776:
3764:
3762:
3761:
3756:
3735:Mantel's theorem
3718:
3716:
3715:
3710:
3708:
3707:
3681:
3679:
3678:
3673:
3661:
3659:
3658:
3653:
3641:
3639:
3638:
3633:
3621:
3619:
3618:
3613:
3595:
3593:
3592:
3587:
3569:
3567:
3566:
3561:
3543:
3541:
3540:
3535:
3508:
3506:
3505:
3500:
3488:
3486:
3485:
3480:
3468:
3466:
3465:
3460:
3448:
3446:
3445:
3440:
3428:
3426:
3425:
3420:
3398:
3396:
3395:
3390:
3388:
3387:
3362:
3360:
3359:
3354:
3352:
3351:
3307:complement graph
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3263:
3261:
3247:
3238:
3236:
3235:
3230:
3211:
3209:
3208:
3203:
3201:
3200:
3170:
3168:
3167:
3162:
3157:
3155:
3148:
3147:
3134:
3123:
3121:
3114:
3113:
3100:
3095:
3093:
3086:
3085:
3072:
3058:of size at least
3053:
3051:
3050:
3045:
3043:
3042:
3024:
3023:
3011:
3010:
2974:
2972:
2971:
2966:
2964:
2962:
2961:
2952:
2951:
2943:
2937:
2928:
2926:
2925:
2920:
2908:
2906:
2905:
2900:
2898:
2890:
2885:
2884:
2868:
2866:
2865:
2860:
2858:
2854:
2853:
2845:
2832:
2824:
2807:
2805:
2804:
2799:
2787:
2785:
2784:
2779:
2774:
2773:
2761:
2760:
2735:
2733:
2732:
2727:
2716:
2715:
2703:
2702:
2683:
2681:
2680:
2675:
2670:
2669:
2657:
2656:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2604:
2603:
2579:
2578:
2554:
2553:
2535:
2534:
2512:
2510:
2509:
2504:
2486:
2484:
2483:
2478:
2476:
2475:
2463:
2462:
2439:
2437:
2436:
2431:
2419:
2417:
2416:
2411:
2409:
2408:
2390:
2389:
2377:
2376:
2360:
2358:
2357:
2352:
2350:
2349:
2340:
2339:
2329:
2328:
2325:
2321:
2300:
2299:
2281:
2280:
2268:
2267:
2245:
2243:
2242:
2237:
2207:
2205:
2204:
2199:
2197:
2196:
2162:
2160:
2159:
2154:
2152:
2151:
2135:
2133:
2132:
2127:
2125:
2124:
2108:
2106:
2105:
2100:
2082:
2080:
2079:
2074:
2066:
2062:
2061:
2045:
2041:
2040:
2020:
2018:
2017:
2012:
2010:
2009:
1990:
1988:
1987:
1982:
1974:
1970:
1969:
1955:
1939:
1937:
1936:
1931:
1929:
1928:
1923:
1919:
1918:
1903:
1883:
1881:
1880:
1875:
1873:
1868:
1867:
1858:
1856:
1852:
1851:
1843:
1821:
1819:
1818:
1813:
1808:
1804:
1803:
1802:
1797:
1793:
1792:
1777:
1765:
1764:
1750:
1742:
1737:
1733:
1732:
1719:
1715:
1714:
1700:
1676:
1674:
1673:
1668:
1666:
1665:
1647:
1646:
1634:
1633:
1614:
1612:
1611:
1606:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1545:
1544:
1528:
1526:
1525:
1520:
1508:
1506:
1505:
1500:
1498:
1497:
1475:
1473:
1472:
1467:
1455:
1453:
1452:
1447:
1435:
1433:
1432:
1427:
1412:
1410:
1409:
1404:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1352:
1350:
1349:
1344:
1332:
1330:
1329:
1324:
1296:
1294:
1293:
1288:
1255:
1253:
1252:
1247:
1232:
1230:
1229:
1224:
1212:
1210:
1209:
1204:
1192:
1190:
1189:
1184:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1109:
1101:
1071:
1069:
1068:
1063:
1051:
1049:
1048:
1043:
1041:
1040:
1039:
1026:
1006:other vertices.
1005:
1003:
1002:
997:
979:
977:
976:
971:
959:
957:
956:
951:
949:
948:
933:vertices in the
932:
930:
929:
924:
912:
910:
909:
904:
892:
890:
889:
884:
882:
881:
865:
863:
862:
857:
845:
843:
842:
837:
835:
834:
808:
806:
805:
800:
788:
786:
785:
780:
768:
766:
765:
760:
740:
738:
737:
732:
714:
712:
711:
706:
694:
692:
691:
686:
662:
660:
659:
654:
627:
625:
624:
619:
617:
612:
611:
602:
600:
596:
595:
587:
567:
565:
564:
559:
557:
549:
534:
532:
531:
526:
499:
497:
496:
491:
475:
473:
472:
467:
465:
460:
459:
450:
448:
444:
428:
420:
401:, this graph has
400:
398:
397:
392:
380:
378:
377:
372:
345:
343:
342:
337:
335:
334:
312:
310:
309:
304:
292:
290:
289:
284:
263:Mantel's theorem
236:-vertex graphs.
235:
231:
219:
217:
216:
211:
181:
179:
178:
173:
161:
159:
158:
153:
141:
139:
138:
133:
131:
130:
108:
106:
105:
100:
72:
70:
69:
64:
36:undirected graph
6421:
6420:
6416:
6415:
6414:
6412:
6411:
6410:
6391:
6390:
6389:
6388:
6379:
6378:
6374:
6335:
6334:
6330:
6298:
6293:
6292:
6288:
6247:
6246:
6242:
6203:
6187:
6186:
6182:
6158:
6157:
6153:
6139:
6138:
6134:
6086:
6085:
6081:
6058:
6049:
6048:
6044:
6038:
6008:
6007:
5988:
5974:
5973:
5966:
5949:
5948:
5939:
5934:
5921:
5892:
5887:
5886:
5855:
5854:
5830:
5808:
5807:
5784:
5783:
5764:
5763:
5714:
5698:
5697:
5678:
5677:
5652:
5647:
5646:
5614:
5609:
5608:
5587:
5582:
5581:
5551:
5550:
5531:
5530:
5505:
5500:
5499:
5480:
5479:
5469:weighted graphs
5444:
5439:
5438:
5417:
5412:
5411:
5387:
5371:
5370:
5369:strictly above
5363:
5337:
5332:
5331:
5282:
5281:
5232:
5223:
5222:
5191:
5184:
5154:
5153:
5134:
5133:
5112:
5107:
5106:
5072:
5071:
5052:
5051:
5012:
5011:
4990:
4984:
4983:
4949:
4948:
4927:
4922:
4921:
4895:
4894:
4875:
4874:
4837:
4836:
4815:
4810:
4809:
4782:
4777:
4776:
4757:
4756:
4727:
4726:
4705:
4700:
4699:
4672:
4667:
4666:
4663:
4635:
4634:
4604:
4599:
4598:
4579:
4578:
4553:
4552:
4525:
4520:
4519:
4500:
4499:
4450:
4449:
4426:
4425:
4397:
4396:
4370:
4326:
4314:
4310:
4305:
4304:
4285:
4284:
4256:
4255:
4233:
4232:
4204:
4193:
4192:
4162:
4123:
4119:
4114:
4113:
4112:-free graph is
4086:
4081:
4080:
4077:
4072:
4070:Generalizations
4035:
4027:
4026:
3987:
3979:
3978:
3959:
3958:
3908:
3903:
3902:
3870:
3865:
3864:
3840:
3835:
3834:
3799:
3791:
3790:
3767:
3766:
3741:
3740:
3737:
3693:
3688:
3687:
3664:
3663:
3662:with copies of
3644:
3643:
3624:
3623:
3598:
3597:
3572:
3571:
3546:
3545:
3514:
3513:
3491:
3490:
3489:with a copy of
3471:
3470:
3451:
3450:
3431:
3430:
3405:
3404:
3373:
3368:
3367:
3337:
3332:
3331:
3327:
3276:
3275:
3251:
3241:
3240:
3221:
3220:
3186:
3181:
3180:
3139:
3138:
3105:
3104:
3077:
3076:
3060:
3059:
3056:independent set
3034:
3015:
3002:
2997:
2996:
2981:
2953:
2938:
2931:
2930:
2911:
2910:
2876:
2871:
2870:
2837:
2833:
2818:
2817:
2811:
2790:
2789:
2765:
2752:
2738:
2737:
2707:
2694:
2686:
2685:
2661:
2648:
2640:
2639:
2620:
2619:
2595:
2570:
2545:
2526:
2515:
2514:
2489:
2488:
2467:
2454:
2449:
2448:
2422:
2421:
2400:
2381:
2368:
2363:
2362:
2341:
2331:
2291:
2272:
2259:
2248:
2247:
2210:
2209:
2182:
2177:
2176:
2169:
2143:
2138:
2137:
2116:
2111:
2110:
2085:
2084:
2053:
2049:
2032:
2028:
2023:
2022:
2001:
1996:
1995:
1961:
1957:
1942:
1941:
1910:
1906:
1905:
1890:
1889:
1859:
1835:
1831:
1826:
1825:
1784:
1780:
1779:
1756:
1755:
1751:
1724:
1720:
1706:
1702:
1681:
1680:
1657:
1638:
1625:
1620:
1619:
1597:
1596:
1589:
1558:
1557:
1536:
1531:
1530:
1511:
1510:
1483:
1478:
1477:
1458:
1457:
1438:
1437:
1418:
1417:
1395:
1394:
1375:
1374:
1355:
1354:
1335:
1334:
1315:
1314:
1307:
1258:
1257:
1238:
1237:
1215:
1214:
1195:
1194:
1175:
1174:
1155:
1154:
1077:
1076:
1054:
1053:
1021:
1015:
1014:
982:
981:
962:
961:
940:
935:
934:
915:
914:
895:
894:
873:
868:
867:
848:
847:
846:-free graph on
820:
815:
814:
791:
790:
771:
770:
751:
750:
717:
716:
697:
696:
677:
676:
669:
645:
644:
634:
603:
579:
575:
570:
569:
537:
536:
502:
501:
482:
481:
451:
412:
408:
403:
402:
383:
382:
348:
347:
320:
315:
314:
295:
294:
275:
274:
271:
233:
230:
221:
187:
186:
164:
163:
144:
143:
116:
111:
110:
109:-vertex clique
79:
78:
55:
54:
32:Turán's theorem
24:
17:
12:
11:
5:
6419:
6417:
6409:
6408:
6403:
6393:
6392:
6387:
6386:
6372:
6344:(3): 683–707,
6328:
6309:(4): 603–618.
6286:
6240:
6214:(1): 106–112,
6193:Goodman, A. W.
6180:
6151:
6132:
6088:Motzkin, T. S.
6079:
6042:
6036:
6010:Aigner, Martin
5986:
5964:
5936:
5935:
5933:
5930:
5929:
5928:
5920:
5917:
5899:
5895:
5874:
5871:
5868:
5865:
5862:
5839:
5836:
5833:
5829:
5824:
5821:
5818:
5815:
5791:
5771:
5749:
5746:
5741:
5738:
5735:
5732:
5729:
5723:
5720:
5717:
5713:
5708:
5705:
5685:
5659:
5655:
5634:
5629:
5625:
5621:
5617:
5594:
5590:
5565:
5560:
5538:
5512:
5508:
5487:
5451:
5447:
5424:
5420:
5396:
5393:
5390:
5386:
5381:
5378:
5362:
5359:
5344:
5340:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5289:
5266:
5261:
5256:
5250:
5247:
5244:
5241:
5238:
5235:
5231:
5226:
5218:
5213:
5209:
5206:
5203:
5200:
5197:
5194:
5188:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5141:
5119:
5115:
5094:
5091:
5088:
5085:
5082:
5079:
5059:
5033:
5028:
5023:
5020:
5015:
5007:
5002:
4999:
4994:
4971:
4968:
4965:
4962:
4959:
4956:
4934:
4930:
4907:
4904:
4882:
4859:
4856:
4853:
4850:
4847:
4844:
4822:
4818:
4795:
4792:
4789:
4785:
4764:
4740:
4737:
4734:
4712:
4708:
4685:
4682:
4679:
4675:
4662:
4659:
4642:
4617:
4614:
4611:
4607:
4586:
4566:
4563:
4560:
4538:
4535:
4532:
4528:
4507:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4433:
4413:
4410:
4407:
4404:
4382:
4377:
4373:
4366:
4362:
4359:
4356:
4353:
4350:
4344:
4341:
4338:
4335:
4332:
4329:
4325:
4320:
4317:
4313:
4292:
4272:
4269:
4266:
4263:
4240:
4216:
4211:
4207:
4203:
4200:
4174:
4169:
4165:
4158:
4154:
4151:
4148:
4145:
4142:
4137:
4134:
4129:
4126:
4122:
4099:
4096:
4093:
4089:
4076:
4073:
4071:
4068:
4055:
4052:
4048:
4042:
4038:
4034:
4007:
4004:
4000:
3994:
3990:
3986:
3966:
3950:or it must be
3937:
3933:
3929:
3926:
3923:
3919:
3915:
3911:
3887:
3883:
3877:
3873:
3847:
3843:
3822:
3819:
3816:
3812:
3806:
3802:
3798:
3774:
3754:
3751:
3748:
3736:
3733:
3706:
3703:
3700:
3696:
3684:
3683:
3671:
3651:
3631:
3611:
3608:
3605:
3585:
3582:
3579:
3559:
3556:
3553:
3533:
3530:
3527:
3524:
3521:
3510:
3498:
3478:
3458:
3438:
3418:
3415:
3412:
3386:
3383:
3380:
3376:
3350:
3347:
3344:
3340:
3326:
3323:
3283:
3260:
3257:
3254:
3250:
3228:
3217:
3216:
3213:
3199:
3196:
3193:
3189:
3160:
3154:
3151:
3146:
3142:
3137:
3132:
3129:
3126:
3120:
3117:
3112:
3108:
3103:
3098:
3092:
3089:
3084:
3080:
3075:
3070:
3067:
3041:
3037:
3033:
3030:
3027:
3022:
3018:
3014:
3009:
3005:
2980:
2977:
2960:
2956:
2950:
2946:
2942:
2918:
2896:
2893:
2888:
2883:
2879:
2869:. Plugging in
2857:
2851:
2848:
2843:
2840:
2836:
2830:
2827:
2797:
2777:
2772:
2768:
2764:
2759:
2755:
2751:
2748:
2745:
2725:
2722:
2719:
2714:
2710:
2706:
2701:
2697:
2693:
2673:
2668:
2664:
2660:
2655:
2651:
2647:
2627:
2607:
2602:
2598:
2594:
2591:
2588:
2585:
2582:
2577:
2573:
2569:
2566:
2563:
2560:
2557:
2552:
2548:
2544:
2541:
2538:
2533:
2529:
2525:
2522:
2502:
2499:
2496:
2474:
2470:
2466:
2461:
2457:
2429:
2407:
2403:
2399:
2396:
2393:
2388:
2384:
2380:
2375:
2371:
2348:
2344:
2338:
2334:
2320:
2317:
2314:
2310:
2306:
2303:
2298:
2294:
2290:
2287:
2284:
2279:
2275:
2271:
2266:
2262:
2258:
2255:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2195:
2192:
2189:
2185:
2168:
2165:
2150:
2146:
2123:
2119:
2098:
2095:
2092:
2072:
2069:
2065:
2060:
2056:
2052:
2048:
2044:
2039:
2035:
2031:
2008:
2004:
1980:
1977:
1973:
1968:
1964:
1960:
1954:
1950:
1927:
1922:
1917:
1913:
1909:
1902:
1898:
1871:
1866:
1862:
1855:
1849:
1846:
1841:
1838:
1834:
1811:
1807:
1801:
1796:
1791:
1787:
1783:
1776:
1772:
1768:
1763:
1759:
1754:
1748:
1745:
1740:
1736:
1731:
1727:
1723:
1718:
1713:
1709:
1705:
1699:
1696:
1693:
1689:
1664:
1660:
1656:
1653:
1650:
1645:
1641:
1637:
1632:
1628:
1604:
1588:
1585:
1565:
1543:
1539:
1518:
1496:
1493:
1490:
1486:
1465:
1445:
1425:
1402:
1382:
1362:
1342:
1322:
1306:
1303:
1299:
1298:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1245:
1234:
1222:
1202:
1182:
1162:
1153:edges between
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1108:
1104:
1100:
1096:
1093:
1090:
1087:
1084:
1073:
1061:
1038:
1033:
1030:
1025:
995:
992:
989:
969:
947:
943:
922:
902:
880:
876:
855:
833:
830:
827:
823:
798:
778:
758:
730:
727:
724:
704:
684:
668:
665:
652:
633:
630:
615:
610:
606:
599:
593:
590:
585:
582:
578:
555:
552:
547:
544:
524:
521:
518:
515:
512:
509:
489:
463:
458:
454:
447:
443:
440:
437:
434:
431:
426:
423:
418:
415:
411:
390:
370:
367:
364:
361:
358:
355:
333:
330:
327:
323:
302:
282:
270:
267:
225:
209:
206:
203:
200:
197:
194:
171:
162:vertices into
151:
129:
126:
123:
119:
98:
95:
92:
89:
86:
62:
21:Turán's method
15:
13:
10:
9:
6:
4:
3:
2:
6418:
6407:
6404:
6402:
6399:
6398:
6396:
6383:
6376:
6373:
6369:
6365:
6361:
6357:
6352:
6347:
6343:
6339:
6332:
6329:
6324:
6320:
6316:
6312:
6308:
6304:
6297:
6290:
6287:
6283:
6279:
6274:
6269:
6264:
6259:
6255:
6251:
6244:
6241:
6237:
6233:
6229:
6225:
6221:
6217:
6213:
6209:
6202:
6198:
6194:
6190:
6184:
6181:
6177:
6173:
6169:
6165:
6161:
6155:
6152:
6147:
6143:
6136:
6133:
6129:
6125:
6121:
6117:
6112:
6107:
6103:
6099:
6098:
6093:
6092:Straus, E. G.
6089:
6083:
6080:
6076:
6072:
6068:
6064:
6056:
6052:
6046:
6043:
6039:
6033:
6029:
6025:
6021:
6020:
6015:
6011:
6005:
6003:
6001:
5999:
5997:
5995:
5993:
5991:
5987:
5982:
5978:
5971:
5969:
5965:
5960:
5956:
5952:
5946:
5944:
5942:
5938:
5931:
5926:
5923:
5922:
5918:
5916:
5913:
5897:
5893:
5869:
5866:
5863:
5837:
5834:
5831:
5827:
5822:
5819:
5816:
5813:
5803:
5789:
5769:
5747:
5744:
5739:
5736:
5733:
5730:
5727:
5721:
5718:
5715:
5711:
5706:
5703:
5683:
5673:
5657:
5653:
5632:
5627:
5623:
5619:
5615:
5592:
5588:
5580:This gives a
5577:
5563:
5558:
5536:
5526:
5510:
5506:
5485:
5476:
5474:
5470:
5465:
5449:
5445:
5422:
5418:
5394:
5391:
5388:
5384:
5379:
5376:
5368:
5360:
5358:
5342:
5338:
5314:
5311:
5305:
5299:
5296:
5293:
5287:
5277:
5264:
5259:
5254:
5248:
5245:
5239:
5233:
5229:
5224:
5211:
5207:
5204:
5198:
5192:
5174:
5168:
5165:
5162:
5139:
5117:
5113:
5092:
5089:
5083:
5077:
5057:
5047:
5031:
5026:
5021:
5018:
5013:
5000:
4997:
4966:
4963:
4960:
4954:
4932:
4928:
4905:
4902:
4880:
4870:
4854:
4851:
4848:
4842:
4820:
4816:
4793:
4790:
4787:
4783:
4762:
4752:
4738:
4735:
4732:
4710:
4706:
4683:
4680:
4677:
4673:
4660:
4658:
4656:
4640:
4631:
4615:
4612:
4609:
4605:
4584:
4564:
4561:
4558:
4536:
4533:
4530:
4526:
4505:
4482:
4479:
4473:
4467:
4464:
4461:
4455:
4445:
4431:
4408:
4402:
4380:
4375:
4371:
4364:
4357:
4351:
4348:
4342:
4339:
4333:
4327:
4323:
4318:
4315:
4311:
4290:
4267:
4261:
4254:
4238:
4228:
4209:
4205:
4198:
4190:
4172:
4167:
4163:
4156:
4149:
4143:
4140:
4135:
4132:
4127:
4124:
4120:
4097:
4094:
4091:
4087:
4074:
4069:
4067:
4050:
4046:
4040:
4036:
4024:
4020:
4002:
3998:
3992:
3988:
3964:
3955:
3953:
3935:
3931:
3927:
3924:
3921:
3917:
3913:
3909:
3901:
3885:
3881:
3875:
3871:
3861:
3845:
3841:
3820:
3814:
3810:
3804:
3800:
3788:
3772:
3752:
3749:
3746:
3732:
3729:
3725:
3720:
3704:
3701:
3698:
3694:
3669:
3649:
3629:
3609:
3606:
3603:
3583:
3580:
3577:
3557:
3554:
3551:
3531:
3528:
3525:
3522:
3519:
3511:
3496:
3476:
3456:
3436:
3416:
3413:
3410:
3402:
3401:
3400:
3384:
3381:
3378:
3374:
3364:
3348:
3345:
3342:
3338:
3324:
3318:
3314:
3312:
3308:
3299:
3295:
3281:
3258:
3255:
3252:
3248:
3226:
3214:
3197:
3194:
3191:
3187:
3178:
3174:
3173:
3172:
3158:
3152:
3149:
3144:
3140:
3135:
3130:
3127:
3124:
3118:
3115:
3110:
3106:
3101:
3096:
3090:
3087:
3082:
3078:
3073:
3068:
3065:
3057:
3039:
3035:
3031:
3028:
3025:
3020:
3016:
3012:
3007:
3003:
2994:
2990:
2986:
2978:
2976:
2958:
2954:
2944:
2916:
2894:
2891:
2886:
2881:
2877:
2855:
2849:
2846:
2841:
2838:
2834:
2828:
2825:
2815:
2809:
2795:
2770:
2766:
2762:
2757:
2753:
2749:
2746:
2720:
2717:
2712:
2708:
2704:
2699:
2695:
2666:
2662:
2658:
2653:
2649:
2625:
2618:is linear in
2600:
2596:
2592:
2589:
2586:
2583:
2580:
2575:
2571:
2567:
2564:
2561:
2558:
2555:
2550:
2546:
2542:
2539:
2536:
2531:
2527:
2520:
2500:
2497:
2494:
2472:
2468:
2464:
2459:
2455:
2445:
2443:
2427:
2405:
2401:
2397:
2394:
2391:
2386:
2382:
2378:
2373:
2369:
2346:
2342:
2336:
2332:
2318:
2315:
2312:
2308:
2304:
2296:
2292:
2288:
2285:
2282:
2277:
2273:
2269:
2264:
2260:
2253:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2193:
2190:
2187:
2183:
2174:
2166:
2164:
2148:
2144:
2121:
2117:
2096:
2093:
2090:
2070:
2067:
2063:
2058:
2054:
2050:
2046:
2042:
2037:
2033:
2029:
2006:
2002:
1992:
1978:
1975:
1971:
1966:
1962:
1958:
1952:
1925:
1920:
1915:
1911:
1907:
1900:
1887:
1869:
1864:
1860:
1853:
1847:
1844:
1839:
1836:
1832:
1822:
1809:
1805:
1799:
1794:
1789:
1785:
1781:
1774:
1770:
1766:
1761:
1757:
1752:
1746:
1743:
1738:
1734:
1729:
1725:
1721:
1716:
1711:
1707:
1703:
1697:
1694:
1691:
1687:
1678:
1662:
1658:
1654:
1651:
1648:
1643:
1639:
1635:
1630:
1626:
1616:
1602:
1594:
1586:
1584:
1582:
1577:
1563:
1541:
1537:
1516:
1494:
1491:
1488:
1484:
1463:
1443:
1423:
1414:
1400:
1380:
1360:
1340:
1320:
1312:
1304:
1302:
1281:
1278:
1275:
1272:
1269:
1263:
1243:
1235:
1220:
1200:
1180:
1160:
1137:
1134:
1131:
1122:
1119:
1116:
1110:
1102:
1091:
1088:
1085:
1074:
1059:
1052:edges within
1031:
1028:
1012:
1011:
1010:
1007:
993:
990:
987:
967:
945:
941:
920:
900:
878:
874:
853:
831:
828:
825:
821:
796:
776:
756:
747:
728:
725:
722:
702:
682:
673:
666:
664:
650:
642:
638:
631:
629:
613:
608:
604:
597:
591:
588:
583:
580:
576:
553:
550:
545:
542:
519:
516:
513:
507:
487:
479:
476:edges, using
461:
456:
452:
445:
438:
432:
429:
424:
421:
416:
413:
409:
388:
365:
362:
359:
353:
331:
328:
325:
321:
300:
280:
268:
266:
264:
260:
256:
253:in 1941. The
252:
249:
248:mathematician
246:
242:
237:
228:
224:
204:
201:
198:
192:
185:
169:
149:
127:
124:
121:
117:
93:
90:
87:
76:
60:
51:
49:
45:
41:
37:
33:
29:
22:
6381:
6375:
6341:
6337:
6331:
6306:
6302:
6289:
6253:
6249:
6243:
6211:
6207:
6183:
6170:(1): 80–84,
6167:
6163:
6160:Bondy, J. A.
6154:
6145:
6141:
6135:
6101:
6095:
6082:
6066:
6062:
6045:
6018:
5980:
5976:
5958:
5954:
5914:
5805:
5675:
5579:
5528:
5477:
5466:
5364:
5279:
5049:
4872:
4754:
4664:
4632:
4447:
4230:
4078:
3956:
3862:
3738:
3721:
3685:
3365:
3328:
3304:
3218:
2992:
2989:Joel Spencer
2982:
2810:
2684:with either
2446:
2170:
1993:
1823:
1679:
1617:
1590:
1578:
1509:-free. Now,
1415:
1373:and the set
1308:
1300:
1008:
960:and the set
812:
635:
272:
262:
261:is known as
255:special case
241:Turán graphs
238:
226:
222:
52:
31:
28:graph theory
25:
6256:: 146–172,
6197:Pósa, Louis
6189:Erdős, Paul
6104:: 533–540,
6069:: 249–251,
5951:Turán, Paul
5607:density of
3212:-free graph
3175:Consider a
1581:Turán graph
184:Turán graph
6395:Categories
6051:Erdős, Pál
5932:References
4982:is around
4395:where the
3469:, replace
3313:for more.
2442:Lagrangian
2167:Lagrangian
1311:Paul Erdős
6351:1212.2454
6263:1409.4192
6148:: 163–188
6128:121387797
5961:: 436–452
5867:−
5835:−
5823:−
5817:≤
5762:. Take a
5740:−
5734:≤
5719:−
5707:−
5392:−
5380:−
5312:−
5300:χ
5246:−
5234:χ
5205:−
5193:χ
5078:χ
4480:−
4468:χ
4340:−
4328:χ
4319:−
4262:χ
4128:−
4054:⌋
4033:⌊
4006:⌋
3985:⌊
3952:pancyclic
3818:⌋
3797:⌊
3128:⋯
3029:…
2985:Noga Alon
2842:−
2812:Now, the
2590:…
2565:…
2556:−
2540:…
2420:with sum
2395:…
2309:∑
2286:…
2228:…
2094:≠
2083:for some
2047:≥
1949:∑
1897:∑
1840:−
1771:∑
1767:−
1695:≠
1688:∑
1652:…
1273:−
1135:−
1120:−
1089:−
991:−
667:Induction
584:−
546:−
417:−
269:Statement
251:Pál Turán
245:Hungarian
6368:59321123
6323:26524353
6199:(1966),
6142:Mat. Sb.
6053:(1970),
5919:See also
5473:graphons
3785:-vertex
2909:for all
2326:adjacent
6282:5552776
6228:0186575
6120:0175813
6075:0307975
5983:: 60–61
4653:is the
4019:cliques
3054:has an
1888:to the
980:of the
913:of the
6366:
6321:
6280:
6236:646660
6234:
6226:
6126:
6118:
6073:
6034:
4187:. The
3726:. The
2322:
632:Proofs
232:-free
75:vertex
6364:S2CID
6346:arXiv
6319:S2CID
6299:(PDF)
6278:S2CID
6258:arXiv
6232:S2CID
6204:(PDF)
6124:S2CID
6059:(PDF)
4947:s in
4597:is a
293:with
6032:ISBN
5806:For
5728:<
5090:>
3642:and
3570:and
2987:and
1618:Let
1456:and
1173:and
789:and
715:for
695:and
6356:doi
6342:184
6311:doi
6268:doi
6254:121
6216:doi
6172:doi
6106:doi
6024:doi
5912:s.
5471:or
5464:s?
5152:is
3789:is
3512:If
3403:If
2736:or
2136:to
1529:is
26:In
6397::
6362:,
6354:,
6340:,
6317:.
6307:17
6305:.
6301:.
6276:,
6266:,
6252:,
6230:,
6224:MR
6222:,
6212:18
6210:,
6206:,
6195:;
6191:;
6168:11
6166:,
6146:24
6122:,
6116:MR
6114:,
6102:17
6100:,
6090:;
6071:MR
6067:21
6061:,
6030:,
6012:;
5989:^
5981:10
5979:,
5967:^
5959:48
5940:^
5357:.
4657:.
4630:.
4444:.
4066:.
1991:.
1576:.
1413:.
628:.
229:+1
30:,
6358::
6348::
6313::
6270::
6260::
6218::
6174::
6108::
6026::
5898:r
5894:K
5873:)
5870:1
5864:r
5861:(
5838:1
5832:r
5828:1
5820:1
5814:d
5802:.
5790:d
5770:t
5748:t
5745:1
5737:1
5731:d
5722:1
5716:t
5712:1
5704:1
5684:t
5658:r
5654:K
5633:.
5628:2
5624:/
5620:k
5616:d
5593:r
5589:K
5564:N
5559:d
5537:N
5511:r
5507:K
5486:d
5450:r
5446:K
5423:r
5419:K
5395:1
5389:r
5385:1
5377:1
5343:a
5339:K
5318:)
5315:1
5309:)
5306:H
5303:(
5297:,
5294:n
5291:(
5288:T
5265:.
5260:a
5255:)
5249:1
5243:)
5240:H
5237:(
5230:n
5225:(
5217:)
5212:a
5208:1
5202:)
5199:H
5196:(
5187:(
5181:)
5178:)
5175:1
5172:(
5169:o
5166:+
5163:1
5160:(
5140:H
5118:a
5114:K
5093:a
5087:)
5084:H
5081:(
5058:H
5032:a
5027:)
5022:r
5019:n
5014:(
5006:)
5001:a
4998:r
4993:(
4970:)
4967:r
4964:,
4961:n
4958:(
4955:T
4933:a
4929:K
4906:r
4903:n
4881:r
4858:)
4855:r
4852:,
4849:n
4846:(
4843:T
4821:a
4817:K
4794:1
4791:+
4788:r
4784:K
4763:n
4739:2
4736:=
4733:a
4711:a
4707:K
4684:1
4681:+
4678:r
4674:K
4641:H
4616:1
4613:+
4610:r
4606:K
4585:H
4565:1
4562:+
4559:r
4537:1
4534:+
4531:r
4527:K
4506:H
4486:)
4483:1
4477:)
4474:H
4471:(
4465:,
4462:n
4459:(
4456:T
4432:H
4412:)
4409:1
4406:(
4403:o
4381:2
4376:2
4372:n
4365:)
4361:)
4358:1
4355:(
4352:o
4349:+
4343:1
4337:)
4334:H
4331:(
4324:1
4316:1
4312:(
4291:H
4271:)
4268:H
4265:(
4239:H
4215:)
4210:2
4206:n
4202:(
4199:o
4173:2
4168:2
4164:n
4157:)
4153:)
4150:1
4147:(
4144:o
4141:+
4136:r
4133:1
4125:1
4121:(
4098:1
4095:+
4092:r
4088:K
4051:4
4047:/
4041:2
4037:n
4003:4
3999:/
3993:2
3989:n
3965:n
3936:2
3932:/
3928:n
3925:,
3922:2
3918:/
3914:n
3910:K
3886:4
3882:/
3876:2
3872:n
3846:n
3842:K
3821:.
3815:4
3811:/
3805:2
3801:n
3773:n
3753:2
3750:=
3747:r
3705:1
3702:+
3699:r
3695:K
3682:.
3670:v
3650:w
3630:u
3610:w
3607:,
3604:u
3584:w
3581:,
3578:v
3558:v
3555:,
3552:u
3532:w
3529:,
3526:v
3523:,
3520:u
3497:u
3477:v
3457:v
3437:u
3417:v
3414:,
3411:u
3385:1
3382:+
3379:r
3375:K
3349:1
3346:+
3343:r
3339:K
3282:S
3259:1
3256:+
3253:d
3249:1
3227:d
3198:1
3195:+
3192:r
3188:K
3159:.
3153:1
3150:+
3145:n
3141:d
3136:1
3131:+
3125:+
3119:1
3116:+
3111:2
3107:d
3102:1
3097:+
3091:1
3088:+
3083:1
3079:d
3074:1
3069:=
3066:S
3040:n
3036:d
3032:,
3026:,
3021:2
3017:d
3013:,
3008:1
3004:d
2959:2
2955:n
2949:|
2945:E
2941:|
2917:i
2895:n
2892:1
2887:=
2882:i
2878:x
2856:)
2850:r
2847:1
2839:1
2835:(
2829:2
2826:1
2796:r
2776:)
2771:j
2767:x
2763:+
2758:i
2754:x
2750:,
2747:0
2744:(
2724:)
2721:0
2718:,
2713:j
2709:x
2705:+
2700:i
2696:x
2692:(
2672:)
2667:j
2663:x
2659:,
2654:i
2650:x
2646:(
2626:t
2606:)
2601:n
2597:x
2593:,
2587:,
2584:t
2581:+
2576:j
2572:x
2568:,
2562:,
2559:t
2551:i
2547:x
2543:,
2537:,
2532:1
2528:x
2524:(
2521:f
2501:j
2498:,
2495:i
2473:j
2469:x
2465:,
2460:i
2456:x
2428:1
2406:n
2402:x
2398:,
2392:,
2387:2
2383:x
2379:,
2374:1
2370:x
2347:j
2343:x
2337:i
2333:x
2319:j
2316:,
2313:i
2305:=
2302:)
2297:n
2293:x
2289:,
2283:,
2278:2
2274:x
2270:,
2265:1
2261:x
2257:(
2254:f
2234:n
2231:,
2225:,
2222:2
2219:,
2216:1
2194:1
2191:+
2188:r
2184:K
2149:i
2145:S
2122:j
2118:S
2097:j
2091:i
2071:2
2068:+
2064:|
2059:j
2055:S
2051:|
2043:|
2038:i
2034:S
2030:|
2007:i
2003:S
1979:n
1976:=
1972:|
1967:i
1963:S
1959:|
1953:i
1926:2
1921:|
1916:i
1912:S
1908:|
1901:i
1870:2
1865:2
1861:n
1854:)
1848:r
1845:1
1837:1
1833:(
1810:,
1806:)
1800:2
1795:|
1790:i
1786:S
1782:|
1775:i
1762:2
1758:n
1753:(
1747:2
1744:1
1739:=
1735:|
1730:j
1726:S
1722:|
1717:|
1712:i
1708:S
1704:|
1698:j
1692:i
1663:r
1659:S
1655:,
1649:,
1644:2
1640:S
1636:,
1631:1
1627:S
1603:r
1564:B
1542:r
1538:K
1517:B
1495:1
1492:+
1489:r
1485:K
1464:B
1444:A
1424:A
1401:v
1381:B
1361:v
1341:A
1321:v
1285:)
1282:r
1279:,
1276:r
1270:n
1267:(
1264:T
1244:B
1233:.
1221:A
1201:B
1181:B
1161:A
1141:)
1138:r
1132:n
1129:(
1126:)
1123:1
1117:r
1114:(
1111:=
1107:|
1103:B
1099:|
1095:)
1092:1
1086:r
1083:(
1072:.
1060:A
1037:)
1032:2
1029:r
1024:(
994:r
988:n
968:B
946:r
942:K
921:r
901:A
879:r
875:K
854:n
832:1
829:+
826:r
822:K
809:.
797:B
777:A
757:A
741:.
729:3
726:=
723:r
703:B
683:A
651:r
614:2
609:2
605:n
598:)
592:r
589:1
581:1
577:(
554:r
551:1
543:1
523:)
520:r
517:,
514:n
511:(
508:T
488:n
462:2
457:2
453:n
446:)
442:)
439:1
436:(
433:o
430:+
425:r
422:1
414:1
410:(
389:r
369:)
366:r
363:,
360:n
357:(
354:T
332:1
329:+
326:r
322:K
301:n
281:G
234:n
227:r
223:K
208:)
205:r
202:,
199:n
196:(
193:T
170:r
150:n
128:1
125:+
122:r
118:K
97:)
94:1
91:+
88:r
85:(
73:-
61:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
