370:
188:
34:
738:
There is also the question of whether any sufficiently large set of points in general position has an "empty" convex quadrilateral, pentagon, etc., that is, one that contains no other input point. The original solution to the happy ending problem can be adapted to show that any five points in general
739:
position have an empty convex quadrilateral, as shown in the illustration, and any ten points in general position have an empty convex pentagon. However, there exist arbitrarily large sets of points in general position that contain no empty convex
443:
729:
649:
758:
proved that every sufficiently large point set in general position contains a convex empty hexagon. More specifically, Gerken showed that the number of points needed is no more than
840:
in sufficiently large planar point sets, by projecting the higher-dimensional point set into an arbitrary two-dimensional subspace. However, the number of points necessary to find
566:
982:. They carried out a computer search which eliminated all possible configurations of 17 points without convex hexagons while examining only a tiny fraction of all configurations.
500:
177:
1770:
1454:
1103:, Ex. 7.3.6, p. 126. This result follows by applying a Ramsey-theoretic argument similar to Szekeres's original proof together with Perles's result on the case
138:
1408:
782:(9). At least 30 points are needed; there exists a set of 29 points in general position with no empty convex hexagon. The question was finally answered by
1285:
Heule, Marijn J. H.; Scheucher, Manfred (2024), "Happy Ending: An Empty
Hexagon in Every Set of 30 Points", in Finkbeiner, Bernd; Kovács, Laura (eds.),
848:
may be smaller in higher dimensions than it is in the plane, and it is possible to find subsets that are more highly constrained. In particular, in
1742:
Surveys on
Discrete and Computational Geometry: Twenty Years Later: AMS-IMS-SIAM Joint Summer Research Conference, June 18-22, 2006, Snowbird, Utah
379:
116:
states precisely a more general relationship between the number of points in a general-position point set and its largest subset forming a convex
923:
292:; the more difficult part of the proof is to show that every set of nine points in general position contains the vertices of a convex pentagon.
1766:
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1709:, Mathematical Sciences Research Institute Publications, vol. 52, Cambridge University Press, pp. 557–568, archived from
1701:
1626:
573:
1531:
786:, who showed, using a SAT solving approach, that indeed every set of 30 points in general position contains an empty hexagon.
1828:
1554:(1994), "The rectilinear crossing number of a complete graph and Sylvester's "four point problem" of geometric probability",
1357:
799:
224:
27:
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511:
448:
37:
The happy ending problem: every set of five points in general position contains the vertices of a convex quadrilateral
1018:
1808:
120:, namely that the smallest number of points for which any general position arrangement contains a convex subset of
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931:
93:
The happy ending theorem can be proven by a simple case analysis: if four or more points are vertices of the
1818:
1838:
1833:
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20:
943:
In this context, general position means that no two points coincide and no three points are collinear.
1388:
1245:
1006:
1452:
Morris, W.; Soltan, V. (2000), "The Erdős-Szekeres problem on points in convex position—A survey",
905:
1091:, Ex. 6.5.6, p.120. GrĂĽnbaum attributes this result to a private communication of Micha A. Perles.
1736:
1611:
1593:
1573:
1376:
1326:
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101:
with two points inside it, the two inner points and one of the triangle sides can be chosen. See
1395:, Congressus Numerantium, vol. 1, Baton Rouge, La.: Louisiana State Univ., pp. 180–188
1237:
1688:
Tóth, G.; Valtr, P. (2005), "The Erdős-Szekeres theorem: upper bounds and related results", in
215:, any sufficiently large finite set of points in the plane in general position has a subset of
143:
1823:
1781:
1746:
1259:
97:, any four such points can be chosen. If on the other hand, the convex hull has the form of a
26:"Erdős–Szekeres conjecture" redirects here. For their theorem on monotonic subsequences, see
1745:, Contemporary Mathematics, vol. 453, American Mathematical Society, pp. 433–442,
1728:
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1289:, Lecture Notes in Computer Science, vol. 14570, Springer-Verlag, pp. 61–80,
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points minimizing the number of convex quadrilaterals is equivalent to minimizing the
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1500:
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defined above, while Nicolás showed that the number of points needed is no more than
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187:
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33:
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supplies a simplification of Gerken's proof that however requires more points,
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1519:
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1229:
833:
greater than the dimension: this follows immediately from existence of convex
363:
1790:
1697:
1125:
1371:
965:, this was first proved by E. Makai; the first published proof appeared in
810:. The number of quadrilaterals must be proportional to the fourth power of
740:
282:
98:
1340:
1317:; Tardos, Gábor (2020), "Two extensions of the Erdős–Szekeres problem",
369:
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1148:
747:
209:
117:
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1391:; Stanton, R.G. (1970), "A combinatorial problem on convex regions",
1213:
Gerken, Tobias (2008), "Empty convex hexagons in planar point sets",
1584:
Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem",
1569:
86:
This was one of the original results that led to the development of
1598:
1331:
1295:
1664:
Tóth, G.; Valtr, P. (1998), "Note on the Erdős-Szekeres theorem",
1272:
Harborth, Heiko (1978), "Konvexe FĂĽnfecke in ebenen
Punktmengen",
373:
A set of sixteen points in general position with no convex hexagon
368:
186:
1287:
Tools and
Algorithms for the Construction and Analysis of Systems
191:
A set of eight points in general position with no convex pentagon
16:
Five coplanar points have a subset forming a convex quadrilateral
1393:
Proc. Louisiana Conf. Combinatorics, Graph Theory and
Computing
1254:, Graduate Texts in Mathematics, vol. 221 (2nd ed.),
1503:(2003), "Finding sets of points without empty convex 6-gons",
966:
438:{\displaystyle f(N)=1+2^{N-2}\quad {\text{for all }}N\geq 3.}
1189:(1961), "On some extremum problems in elementary geometry",
856: + 3 points in general position have a subset of
445:
They proved later, by constructing explicit examples, that
1627:"Computer solution to the 17-point Erdős-Szekeres problem"
1355:
Horton, J. D. (1983), "Sets with no empty convex 7-gons",
821:, sufficiently large sets of points will have a subset of
817:
It is straightforward to show that, in higher-dimensional
179:. It remains unproven, but less precise bounds are known.
1030:
1476:
Nicolás, Carlos M. (2007), "The empty hexagon theorem",
75:
has a subset of four points that form the vertices of a
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For a long time the question of the existence of empty
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This was the original problem, proved by Esther Klein.
661:
576:
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382:
223:
The proof appeared in the same paper that proves the
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126:
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Ramsey-theoretic proof of the Erdős-Szekeres theorem
1564:(10), Mathematical Association of America: 939–943,
1319:
Journal of the
European Mathematical Society (JEMS)
227:on monotonic subsequences in sequences of numbers.
219:points that form the vertices of a convex polygon.
1313:Holmsen, Andreas F.; Mojarrad, Hossein Nassajian;
860: + 2 points that form the vertices of a
724:{\displaystyle f(N)\leq 2^{N+O({\sqrt {NlogN}})}.}
723:
643:
560:
494:
437:
171:
132:
105:for an illustrated explanation of this proof, and
570:Suk actually proves, for N sufficiently large,
285:is shown in the illustration, demonstrating that
249:points in general position must contain a convex
1409:"Finding convex sets among points in the plane"
1077:
979:
783:
1455:Bulletin of the American Mathematical Society
1132:(1998), "Forced convex n-gons in the plane",
991:
967:Kalbfleisch, Kalbfleisch & Stanton (1970)
962:
900:) points in general position has a subset of
323:
195:
8:
1208:, Cambridge, MA: MIT Press, pp. 680–689
1191:Ann. Univ. Sci. Budapest. Eötvös Sect. Math.
644:{\displaystyle f(N)\leq 2^{N+6N^{2/3}logN}.}
106:
924:A world of teaching and numbers - times two
202:
109:for a more detailed survey of the problem.
65:
1727:Valtr, P. (2008), "On empty hexagons", in
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1518:
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1703:Combinatorial and Computational Geometry
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281:. A set of eight points with no convex
71:any set of five points in the plane in
1206:The Art of Counting: Selected Writings
1054:
755:
1165:"A combinatorial problem in geometry"
771:
337:is known to be finite for all finite
198:proved the following generalisation:
7:
825:points that forms the vertices of a
561:{\displaystyle f(N)\leq 2^{N+o(N)}.}
347:On the basis of the known values of
1667:Discrete and Computational Geometry
1506:Discrete and Computational Geometry
1479:Discrete and Computational Geometry
1417:Discrete and Computational Geometry
1216:Discrete and Computational Geometry
1135:Discrete and Computational Geometry
1002:
904:points that form the vertices of a
653:This was subsequently improved to:
502:In 2016 Andrew Suk showed that for
495:{\displaystyle f(N)\geq 1+2^{N-2}.}
53:because it led to the marriage of
14:
362:= 3, 4 and 5, Erdős and Szekeres
794:The problem of finding sets of
420:
1358:Canadian Mathematical Bulletin
934:, 2005-11-07, cited 2014-09-04
713:
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366:in their original paper that
61:) is the following statement:
19:For the Fred Frith album, see
1:
1557:American Mathematical Monthly
1469:10.1090/S0273-0979-00-00877-6
1078:Scheinerman & Wilf (1994)
1021:for the asymptotic expansion.
980:Szekeres & Peters (2006)
864:. More generally, for every
784:Heule & Scheucher (2024)
1306:10.1007/978-3-031-57246-3_5
1204:(1973), Spencer, J. (ed.),
1013:for notation used here and
992:Erdős & Szekeres (1961)
963:Erdős & Szekeres (1935)
324:Erdős & Szekeres (1935)
196:Erdős & Szekeres (1935)
1855:
762:(9) for the same function
107:Morris & Soltan (2000)
25:
18:
1646:10.1017/S144618110000300X
1520:10.1007/s00454-002-2829-x
1493:10.1007/s00454-007-1343-6
1230:10.1007/s00454-007-9018-x
932:The Sydney Morning Herald
888:) such that every set of
172:{\displaystyle 2^{n-2}+1}
114:Erdős–Szekeres conjecture
1814:Euclidean plane geometry
1240:(2003), Kaibel, Volker;
1019:Stirling's approximation
978:This has been proved by
1274:Elemente der Mathematik
253:-gon. It is known that
1548:Scheinerman, Edward R.
1372:10.4153/CMB-1983-077-8
1170:Compositio Mathematica
876:there exists a number
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225:Erdős–Szekeres theorem
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28:Erdős–Szekeres theorem
1829:Mathematical problems
1625:; Peters, L. (2006),
1031:Holmsen et al. (2020)
734:Empty convex polygons
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245:for which any set of
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21:The Happy End Problem
1767:Happy ending problem
1532:"Planes of Budapest"
1007:binomial coefficient
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47:happy ending problem
1786:"Happy End Problem"
1586:J. Amer. Math. Soc.
1387:Kalbfleisch, J.D.;
906:neighborly polytope
802:in a straight-line
750:remained open, but
322:. By the result of
315:is unknown for all
241:denote the minimum
206: —
69: —
1782:Weisstein, Eric W.
1681:10.1007/PL00009363
1431:10.1007/PL00009358
1246:Ziegler, GĂĽnter M.
1149:10.1007/PL00009353
852:dimensions, every
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1809:Discrete geometry
1729:Goodman, Jacob E.
1690:Goodman, Jacob E.
1389:Kalbfleisch, J.G.
1341:10.4171/jems/1000
1325:(12): 3981–3995,
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208:for any positive
133:{\displaystyle n}
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1737:Pollack, Richard
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1649:, archived from
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1552:Wilf, Herbert S.
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778:(15) instead of
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1632:ANZIAM Journal
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1716:on 2019-07-28
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1039:
1036:
1032:
1027:
1024:
1020:
1016:
1012:
1008:
1004:
999:
996:
993:
988:
985:
981:
975:
972:
968:
964:
961:According to
958:
955:
949:
946:
940:
937:
933:
929:
925:
920:
917:
911:
909:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
851:
847:
843:
837:
832:
828:
824:
820:
815:
813:
809:
805:
801:
797:
789:
787:
785:
781:
777:
773:
769:
765:
761:
757:
756:Gerken (2008)
753:
749:
744:
742:
733:
731:
718:
708:
705:
702:
699:
696:
688:
685:
682:
678:
674:
668:
662:
654:
651:
638:
633:
630:
627:
624:
619:
615:
611:
607:
603:
600:
597:
593:
589:
583:
577:
568:
555:
547:
541:
538:
535:
531:
527:
521:
515:
506:
489:
484:
481:
478:
474:
470:
467:
464:
458:
452:
432:
429:
426:
423:for all
415:
412:
409:
405:
401:
398:
395:
389:
383:
371:
367:
365:
361:
355:
351:
341:
334:
330:
325:
319:
312:
308:
304:The value of
303:
298:
294:
289:
284:
278:
274:
269:
265:
260:
256:
255:
254:
252:
238:
234:
228:
226:
220:
211:
199:
197:
189:
182:
180:
166:
163:
158:
155:
152:
148:
127:
119:
115:
110:
108:
104:
100:
96:
91:
89:
88:Ramsey theory
83:
81:
80:quadrilateral
78:
74:
62:
60:
56:
52:
48:
44:
35:
29:
22:
1789:
1741:
1718:, retrieved
1711:the original
1702:
1671:
1665:
1655:, retrieved
1651:the original
1636:
1630:
1623:Szekeres, G.
1589:
1585:
1561:
1555:
1540:the original
1535:
1510:
1504:
1501:Overmars, M.
1483:
1477:
1459:
1453:
1443:, retrieved
1436:the original
1421:
1415:
1392:
1362:
1356:
1322:
1318:
1286:
1280:(5): 116–118
1277:
1273:
1250:
1242:Klee, Victor
1220:
1214:
1205:
1194:
1190:
1187:Szekeres, G.
1174:
1168:
1161:Szekeres, G.
1139:
1133:
1130:Graham, R.L.
1108:
1104:
1096:
1084:
1073:
1061:
1050:
1038:
1026:
998:
987:
974:
957:
948:
939:
919:
901:
897:
893:
889:
885:
881:
877:
873:
869:
865:
857:
853:
849:
841:
835:
830:
822:
816:
811:
795:
793:
779:
775:
772:Valtr (2008)
767:
763:
759:
745:
737:
655:
652:
569:
504:
376:
359:
353:
349:
346:
339:
332:
328:
317:
310:
306:
296:
287:
276:
267:
263:, trivially.
258:
250:
236:
232:
229:
222:
201:
194:
113:
111:
92:
85:
64:
59:Esther Klein
46:
40:
1733:Pach, János
1694:Pach, János
1405:Pachter, L.
1315:Pach, János
364:conjectured
95:convex hull
43:mathematics
1839:Paul Erdős
1803:Categories
1775:PlanetMath
1720:2015-02-28
1698:Welzl, Emo
1657:2007-01-05
1599:1604.08657
1536:MAA Online
1445:2019-09-23
1332:1710.11415
1296:2403.00737
1118:References
1003:Suk (2016)
844:points in
829:, for any
290:(5) > 8
140:points is
51:Paul Erdős
1791:MathWorld
1381:120267029
1202:Erdős, P.
1183:Erdős, P.
1177:: 463–470
1157:Erdős, P.
675:≤
590:≤
528:≤
482:−
465:≥
430:≥
413:−
156:−
1824:Polygons
1739:(eds.),
1700:(eds.),
1616:15732134
1530:(2000),
1407:(1998),
1248:(eds.),
1163:(1935),
748:hexagons
741:heptagon
299:(6) = 17
283:pentagon
99:triangle
1578:2975158
1349:4176784
1197:: 53–62
804:drawing
279:(5) = 9
270:(4) = 5
261:(3) = 3
210:integer
203:Theorem
118:polygon
66:Theorem
45:, the "
1749:
1614:
1576:
1379:
1347:
1262:
1005:. See
770:(25).
320:> 6
77:convex
1714:(PDF)
1707:(PDF)
1612:S2CID
1594:arXiv
1574:JSTOR
1439:(PDF)
1412:(PDF)
1377:S2CID
1327:arXiv
1291:arXiv
912:Notes
838:-gons
806:of a
1769:and
1747:ISBN
1260:ISBN
1009:and
868:and
754:and
358:for
230:Let
112:The
57:and
1773:on
1676:doi
1641:doi
1604:doi
1566:doi
1562:101
1515:doi
1488:doi
1464:doi
1426:doi
1367:doi
1337:doi
1301:doi
1225:doi
1195:3–4
1144:doi
1017:or
507:≥ 7
41:In
1805::
1788:,
1784:,
1735:;
1731:;
1696:;
1692:;
1672:19
1670:,
1637:48
1635:,
1629:,
1610:,
1602:,
1590:30
1588:,
1572:,
1560:,
1550:;
1534:,
1511:29
1509:,
1484:38
1482:,
1460:37
1458:,
1422:19
1420:,
1414:,
1403:;
1375:,
1363:26
1361:,
1345:MR
1343:,
1335:,
1323:22
1321:,
1299:,
1278:33
1276:,
1258:,
1244:;
1221:39
1219:,
1193:,
1185:;
1173:,
1167:,
1159:;
1140:19
1138:,
1128:;
930:,
926:,
908:.
896:,
884:,
743:.
433:3.
326:,
90:.
82:.
1678::
1643::
1606::
1596::
1568::
1517::
1490::
1466::
1428::
1369::
1339::
1329::
1303::
1293::
1227::
1175:2
1146::
1109:d
1105:k
1068:.
1045:.
1033:.
969:.
902:k
898:k
894:d
892:(
890:m
886:k
882:d
880:(
878:m
874:d
870:k
866:d
858:d
854:d
850:d
842:k
836:k
831:k
823:k
812:n
796:n
780:f
776:f
768:f
764:f
760:f
719:.
714:)
709:N
706:g
703:o
700:l
697:N
692:(
689:O
686:+
683:N
679:2
672:)
669:N
666:(
663:f
639:.
634:N
631:g
628:o
625:l
620:3
616:/
612:2
608:N
604:6
601:+
598:N
594:2
587:)
584:N
581:(
578:f
556:.
551:)
548:N
545:(
542:o
539:+
536:N
532:2
525:)
522:N
519:(
516:f
505:N
490:.
485:2
479:N
475:2
471:+
468:1
462:)
459:N
456:(
453:f
427:N
416:2
410:N
406:2
402:+
399:1
396:=
393:)
390:N
387:(
384:f
360:N
356:)
354:N
352:(
350:f
343:.
340:N
335:)
333:N
331:(
329:f
318:N
313:)
311:N
309:(
307:f
301:.
297:f
288:f
277:f
272:.
268:f
259:f
251:N
247:M
243:M
239:)
237:N
235:(
233:f
217:N
213:N
167:1
164:+
159:2
153:n
149:2
128:n
30:.
23:.
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