43:-dimensional subspaces, or even more generally, objects of type 1 and objects of type 2 when some concept of intersection makes sense for these objects. A second way to generalize would be to move into more abstract settings than projective geometry. One can define a blocking set of a
94:
It is sometimes useful to drop the condition that a blocking set does not contain a line. Under this extended definition, and since, in a projective plane every pair of lines meet, every line would be a blocking set. Blocking sets which contained lines would be called
781:
was defined as a set of points containing no line but intersecting every line. In 1958, J. R. Isbell studied these games from a non-geometric viewpoint. Jane W. DiPaola studied the minimum blocking coalitions in all the projective planes of order
615:
87:. Every committee is a minimal blocking set, but not all minimal blocking sets are committees. Blocking sets exist in all projective planes except for the smallest projective plane of order 2, the
35:
that every line intersects and that does not contain an entire line. The concept can be generalized in several ways. Instead of talking about points and lines, one could deal with
1470:
1680:
1600:
676:
723:
1438:
that intersects every hyperplane non-trivially, i.e., every hyperplane is incident with some point of the set, is called an affine blocking set. Identify the space with
1370:
2094:
1814:
1511:
515:
1436:
1103:
849:
803:
1620:
1571:
1551:
1203:
1183:
1163:
1143:
1123:
1071:
1044:
1024:
1004:
969:
949:
929:
909:
889:
869:
486:
1235:, which can not be extended to a larger arc (thus, every point not on the arc is on a secant line of the arc–a line meeting the arc in two points.)
2138:
2148:
2119:
2048:
1472:
by fixing a coordinate system. Then it is easily shown that the set of points lying on the coordinate axes form a blocking set of size
542:
63:, a blocking set is a set of points of π that every line intersects and that contains no line completely. Under this definition, if
1517:
conference that this is the least possible size of a blocking set. This was proved by R. E. Jamison in 1977, and independently by
740:
is not a square less can be said about the smallest sized nontrivial blocking sets. One well known result due to Aart
Blokhuis is:
2000:
1970:
1944:
1529:. Jamison proved the following general covering result from which the bound on affine blocking sets follows using duality:
1262:
2192:
983:
1218:
437:
are the field elements with absolute trace 0, the condition in the definition of a projective triad is satisfied.
1125:
into two subsets (color classes) such that no edge is monochromatic, i.e., no edge is contained entirely within
2187:
378:
2154:
271:
2177:
1441:
1388:). Not all blocking sets are of Rédei type, but many of the smaller ones are. These sets are named after
1526:
1392:
whose monograph on
Lacunary polynomials over finite fields was influential in the study of these sets.
1389:
1625:
1576:
2182:
645:
695:
2103:
1522:
24:
1833:
521:
2099:
P. Duchet, Hypergraphs, Chapter 7 in: Handbook of
Combinatorics, North-Holland, Amsterdam, 1995.
1340:
682:
is necessarily a square and the blocking set consists of the points in some Baer subplane of π.
119:+ 1 points), the points on the lines forming a triangle without the vertices of the triangle (3(
2144:
2115:
2070:
2054:
2044:
1475:
730:
491:
1403:
1076:
816:
2036:
2009:
1979:
1949:
1923:
1885:
1859:
1823:
1747:
1214:
785:
774:
629:
108:
56:
32:
1876:
DiPaola, Jane W. (1969), "On
Minimum Blocking Coalitions in Small Projective Plane Games",
1518:
1998:
Brouwer, Andries; Schrijver, Alexander (1978), "The blocking number of an affine space",
2108:
1605:
1556:
1536:
1188:
1168:
1148:
1128:
1108:
1056:
1029:
1009:
989:
954:
934:
914:
894:
874:
854:
471:
1954:
468:
One typically searches for small blocking sets. The minimum size of a blocking set of
83:
leaves a set which is not a blocking set. A blocking set of smallest size is called a
2171:
2014:
1984:
1622:-dimensional cosets required to cover all vectors except the zero vector is at least
773:
in a 1956 paper by Moses
Richardson. Players were identified with points in a finite
625:
1914:
Szőnyi, Tamás (1997), "Blocking Sets in
Desarguesian Affine and Projective Planes",
2064:
979:
1863:
228:
of which lie on each of three concurrent lines such that the point of concurrency
1514:
975:
770:
382:
1372:
If for some line equality holds in this relation, the blocking set is called a
2040:
1968:
Jamison, Robert E. (1977), "Covering finite fields with cosets of subspaces",
1050:
88:
44:
2058:
189:
of the triangle are in β, and the following condition is satisfied: If point
1232:
729:
is necessarily a square and the blocking set consists of the points of some
413:). Three points, one from each of these lines, are collinear if and only if
143:
1928:
685:
An upper bound for the size of a minimal blocking set has the same flavor,
326:). Three points, one on each of these sides, are collinear if and only if
350:), the condition in the definition of a projective triangle is satisfied.
20:
138:, on a given line and then one point on each of the other lines through
1837:
1751:
130:
Another general construction in an arbitrary projective plane of order
127:= 2 this blocking set is trivial) which in general is not a committee.
761:
In these planes a projective triangle which meets this bound exists.
1889:
1828:
377:
The construction is similar to the above, but since the field is of
2067:, Graphs and hypergraphs, North-Holland, Amsterdam, 1973. (Defines
1738:
Blokhuis, Aart (1994), "On the size of a blocking set in PG(2,p)",
632:
and the upper bound comes from the complement of a Baer subplane.
610:{\displaystyle q+{\sqrt {q}}+1\leq |B|\leq q^{2}-{\sqrt {q}}.}
381:, squares and non-squares need to be replaced by elements of
232:
is in δ and the following condition is satisfied: If a point
240:
on another line are in δ, then the point of intersection of
688:
Any minimal blocking set in a projective plane π of order
1305:|, the size of the blocking set) consider a line meeting
1812:
Richardson, Moses (1956), "On Finite
Projective Games",
1400:
286:= (0,0,1). The points, other than the vertices, on side
725:
points. Moreover, if this upper bound is reached, then
67:
is a blocking set, then complementary set of points, π\
2073:
1628:
1608:
1579:
1559:
1539:
1478:
1444:
1406:
1343:
1191:
1171:
1151:
1131:
1111:
1079:
1059:
1032:
1012:
992:
957:
937:
917:
897:
877:
857:
819:
788:
698:
648:
545:
494:
474:
971:
that has nonempty intersection with each hyperedge.
769:
Blocking sets originated in the context of economic
150:= 2.) This produces a minimal blocking set of size 2
678:points. Moreover, if this lower bound is met, then
425:. By selecting all the points on these lines where
177:on each side of a triangle, such that the vertices
2107:
2088:
1674:
1614:
1594:
1565:
1545:
1505:
1464:
1430:
1384:will be the largest number of collinear points in
1364:
1197:
1177:
1157:
1137:
1117:
1097:
1065:
1038:
1018:
998:
963:
943:
923:
903:
883:
863:
843:
797:
717:
670:
638:Any blocking set in a projective plane π of order
609:
509:
480:
452:a prime, there exists a projective triad of side (
986:" is used, but in some contexts a transversal of
258:odd, there exists a projective triangle of side (
205:are both in β, then the point of intersection of
47:as a set that meets all edges of the hypergraph.
1942:Szőnyi, Tamás (1999), "Around Rédei's theorem",
1850:Isbell, J.R. (1958), "A Class of Simple Games",
1815:Proceedings of the American Mathematical Society
1046:that meets each hyperedge in exactly one point.
365:even, there exists a projective triad of side (
1733:
1731:
409:have coordinates which may be written as (1,1,
777:and minimal winning coalitions were lines. A
146:(this last condition can not be satisfied if
123:- 1) points) form a minimal blocking set (if
8:
1787:
1775:
1092:
1080:
142:, making sure that these points are not all
2140:Current Research Topics in Galois Geometry
1901:
1763:
1722:
1710:
1698:
385:0 and absolute trace 1. Specifically, let
2072:
2013:
1983:
1953:
1927:
1827:
1633:
1627:
1607:
1586:
1582:
1581:
1578:
1558:
1538:
1477:
1456:
1451:
1447:
1446:
1443:
1405:
1342:
1190:
1170:
1150:
1130:
1110:
1078:
1058:
1031:
1011:
991:
974:Blocking sets are sometimes also called "
956:
936:
916:
911:, called (hyper)edges. A blocking set of
896:
876:
856:
818:
787:
702:
697:
655:
647:
597:
588:
576:
568:
552:
544:
493:
473:
456:+ 1)/2 which is a blocking set of size (3
369:+ 2)/2 which is a blocking set of size (3
262:+ 3)/2 which is a blocking set of size 3(
2110:Projective Geometries over Finite Fields
1333:must each contain at least one point of
628:the lower bound is achieved by any Baer
1691:
71:is also a blocking set. A blocking set
1799:
1313:points. Since no line is contained in
401:= 0 have coordinates of the form (1,0,
393:= 0 have coordinates of the form (0,1,
334:. By choosing all of the points where
274:, let the vertices of the triangle be
2130:Blocking Sets of Conics, Ph.D. thesis
635:A more general result can be proved,
134:is to take all except one point, say
7:
2031:Barwick, Susan; Ebert, Gary (2008),
1916:Finite Fields and Their Applications
1465:{\displaystyle \mathbb {F} _{q}^{n}}
746:: A nontrivial blocking set in PG(2,
322:are elements of the finite field GF(
2137:De Beule, Jan; Storme, Leo (2011),
2114:, Oxford: Oxford University Press,
1878:SIAM Journal on Applied Mathematics
1513:. Jean Doyen conjectured in a 1976
1265:in Π of the set of secant lines of
1293:, for any nontrivial blocking set
14:
1289:In any projective plane of order
1682:. Moreover, this bound is sharp.
1675:{\displaystyle q^{n-k}-1+k(q-1)}
1595:{\displaystyle \mathbb {F} _{q}}
236:on one of the lines and a point
99:blocking sets, in this setting.
2132:, University of Colorado Denver
2001:Journal of Combinatorial Theory
1971:Journal of Combinatorial Theory
1380:of the blocking set (note that
1321:, on this line which is not in
671:{\displaystyle n+{\sqrt {n}}+1}
290:have coordinates of the form (-
79:if the removal of any point of
2083:
2077:
1669:
1657:
1573:dimensional vector space over
1500:
1488:
1425:
1413:
1337:in order to be blocked. Thus,
891:is a collection of subsets of
838:
826:
718:{\displaystyle n{\sqrt {n}}+1}
577:
569:
532:), the size of a blocking set
504:
498:
389:= (0,0,1). Points on the line
16:Concept in projective geometry
1:
1955:10.1016/s0012-365x(99)00097-7
1864:10.1215/s0012-7094-58-02537-7
1525:in 1978 using the so-called
754:a prime, has size at least 3(
522:Desarguesian projective plane
244:with the third line is in δ.
2033:Unitals in Projective Planes
2015:10.1016/0097-3165(78)90013-4
1985:10.1016/0097-3165(77)90001-2
2143:, Nova Science Publishers,
39:-dimensional subspaces and
2209:
2128:Holder, Leanne D. (2001),
1374:blocking set of Rédei type
1365:{\displaystyle b\geq n+q.}
871:is a set of elements, and
346:are nonzero squares of GF(
2041:10.1007/978-0-387-76366-8
1852:Duke Mathematical Journal
1317:, there must be a point,
851:be a hypergraph, so that
397:), and those on the line
220:δ of side m is a set of 3
2089:{\displaystyle \tau (H)}
1904:, p. 366, Theorem 13.1.2
1788:Barwick & Ebert 2008
1776:Barwick & Ebert 2008
1766:, p. 376, Theorem 13.3.3
1725:, p. 377, Theorem 13.4.2
1713:, p. 376, Theorem 13.4.1
1506:{\displaystyle 1+n(q-1)}
510:{\displaystyle \tau (H)}
31:is a set of points in a
1431:{\displaystyle AG(n,q)}
1098:{\displaystyle \{C,D\}}
844:{\displaystyle H=(X,E)}
272:homogeneous coordinates
2090:
2035:, New York: Springer,
1929:10.1006/ffta.1996.0176
1676:
1616:
1596:
1567:
1547:
1507:
1466:
1432:
1366:
1199:
1179:
1159:
1139:
1119:
1099:
1067:
1040:
1020:
1000:
965:
945:
925:
905:
885:
865:
845:
799:
798:{\displaystyle \leq 9}
719:
672:
611:
511:
482:
405:). Points of the line
306:have coordinates (1,0,
298:have coordinates (0,1,
2091:
1790:, p. 30, Theorem 2.16
1778:, p. 30, Theorem 2.15
1677:
1617:
1602:. Then the number of
1597:
1568:
1548:
1508:
1467:
1433:
1367:
1200:
1180:
1160:
1140:
1120:
1100:
1068:
1041:
1021:
1001:
966:
946:
926:
906:
886:
866:
846:
800:
720:
673:
612:
512:
483:
2071:
1948:, 208/209: 557–575,
1945:Discrete Mathematics
1626:
1606:
1577:
1557:
1537:
1476:
1442:
1404:
1396:Affine blocking sets
1341:
1189:
1169:
1149:
1129:
1109:
1077:
1057:
1030:
1010:
990:
955:
935:
915:
895:
875:
855:
817:
786:
696:
646:
543:
492:
472:
115:(each line contains
2193:Projective geometry
1461:
1329:other lines though
1285:Rédei blocking sets
1269:is a blocking set,
1205:are blocking sets.
159:projective triangle
25:projective geometry
2104:Hirschfeld, J.W.P.
2086:
1752:10.1007/bf01305953
1672:
1612:
1592:
1563:
1543:
1503:
1462:
1445:
1428:
1362:
1195:
1175:
1155:
1135:
1115:
1095:
1063:
1036:
1016:
996:
982:". Also the term "
961:
941:
921:
901:
881:
861:
841:
795:
779:blocking coalition
715:
668:
607:
507:
478:
294:, 1, 0), those on
2150:978-1-61209-523-3
2121:978-0-19-853526-3
2050:978-0-387-76364-4
1615:{\displaystyle k}
1566:{\displaystyle n}
1546:{\displaystyle V}
1527:polynomial method
1249:-arc in Π = PG(2,
1231:points, no three
1198:{\displaystyle D}
1178:{\displaystyle C}
1158:{\displaystyle D}
1138:{\displaystyle C}
1118:{\displaystyle X}
1066:{\displaystyle H}
1039:{\displaystyle X}
1019:{\displaystyle T}
999:{\displaystyle H}
964:{\displaystyle X}
944:{\displaystyle S}
924:{\displaystyle H}
904:{\displaystyle X}
884:{\displaystyle E}
864:{\displaystyle X}
707:
660:
602:
557:
481:{\displaystyle H}
2200:
2164:
2163:
2162:
2153:, archived from
2133:
2124:
2113:
2095:
2093:
2092:
2087:
2061:
2019:
2018:
2017:
1995:
1989:
1988:
1987:
1965:
1959:
1958:
1957:
1939:
1933:
1932:
1931:
1911:
1905:
1899:
1893:
1892:
1873:
1867:
1866:
1847:
1841:
1840:
1831:
1809:
1803:
1797:
1791:
1785:
1779:
1773:
1767:
1761:
1755:
1754:
1735:
1726:
1720:
1714:
1708:
1702:
1696:
1681:
1679:
1678:
1673:
1644:
1643:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1591:
1590:
1585:
1572:
1570:
1569:
1564:
1552:
1550:
1549:
1544:
1512:
1510:
1509:
1504:
1471:
1469:
1468:
1463:
1460:
1455:
1450:
1437:
1435:
1434:
1429:
1371:
1369:
1368:
1363:
1215:projective plane
1204:
1202:
1201:
1196:
1184:
1182:
1181:
1176:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1072:
1070:
1069:
1064:
1045:
1043:
1042:
1037:
1025:
1023:
1022:
1017:
1005:
1003:
1002:
997:
970:
968:
967:
962:
950:
948:
947:
942:
930:
928:
927:
922:
910:
908:
907:
902:
890:
888:
887:
882:
870:
868:
867:
862:
850:
848:
847:
842:
804:
802:
801:
796:
775:projective plane
724:
722:
721:
716:
708:
703:
677:
675:
674:
669:
661:
656:
616:
614:
613:
608:
603:
598:
593:
592:
580:
572:
558:
553:
516:
514:
513:
508:
487:
485:
484:
479:
379:characteristic 2
218:projective triad
169:) consists of 3(
109:projective plane
57:projective plane
33:projective plane
2208:
2207:
2203:
2202:
2201:
2199:
2198:
2197:
2188:Finite geometry
2168:
2167:
2160:
2158:
2151:
2136:
2127:
2122:
2102:
2069:
2068:
2051:
2030:
2027:
2022:
1997:
1996:
1992:
1967:
1966:
1962:
1941:
1940:
1936:
1913:
1912:
1908:
1902:Hirschfeld 1979
1900:
1896:
1890:10.1137/0117036
1875:
1874:
1870:
1849:
1848:
1844:
1829:10.2307/2032754
1811:
1810:
1806:
1798:
1794:
1786:
1782:
1774:
1770:
1764:Hirschfeld 1979
1762:
1758:
1737:
1736:
1729:
1723:Hirschfeld 1979
1721:
1717:
1711:Hirschfeld 1979
1709:
1705:
1699:Hirschfeld 1979
1697:
1693:
1689:
1629:
1624:
1623:
1604:
1603:
1580:
1575:
1574:
1555:
1554:
1535:
1534:
1474:
1473:
1440:
1439:
1402:
1401:
1398:
1376:and the line a
1339:
1338:
1287:
1211:
1209:Complete k-arcs
1187:
1186:
1167:
1166:
1147:
1146:
1127:
1126:
1107:
1106:
1075:
1074:
1073:is a partition
1055:
1054:
1028:
1027:
1008:
1007:
988:
987:
953:
952:
933:
932:
913:
912:
893:
892:
873:
872:
853:
852:
815:
814:
811:
784:
783:
767:
733:embedded in π.
694:
693:
644:
643:
584:
541:
540:
490:
489:
470:
469:
466:
302:) and those on
105:
53:
23:, specifically
17:
12:
11:
5:
2206:
2204:
2196:
2195:
2190:
2185:
2180:
2170:
2169:
2166:
2165:
2149:
2134:
2125:
2120:
2100:
2097:
2085:
2082:
2079:
2076:
2062:
2049:
2026:
2023:
2021:
2020:
2008:(2): 251–253,
1990:
1978:(3): 253–266,
1960:
1934:
1922:(3): 187–202,
1906:
1894:
1884:(2): 378–392,
1868:
1858:(3): 425–436,
1842:
1822:(3): 458–465,
1804:
1792:
1780:
1768:
1756:
1727:
1715:
1703:
1690:
1688:
1685:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1642:
1639:
1636:
1632:
1611:
1589:
1584:
1562:
1542:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1459:
1454:
1449:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1397:
1394:
1361:
1358:
1355:
1352:
1349:
1346:
1286:
1283:
1245:be a complete
1210:
1207:
1194:
1174:
1154:
1134:
1114:
1094:
1091:
1088:
1085:
1082:
1062:
1035:
1015:
995:
960:
940:
920:
900:
880:
860:
840:
837:
834:
831:
828:
825:
822:
810:
809:In hypergraphs
807:
794:
791:
766:
763:
714:
711:
706:
701:
667:
664:
659:
654:
651:
618:
617:
606:
601:
596:
591:
587:
583:
579:
575:
571:
567:
564:
561:
556:
551:
548:
506:
503:
500:
497:
477:
465:
462:
439:
438:
383:absolute trace
352:
351:
282:= (0,1,0) and
104:
101:
52:
49:
15:
13:
10:
9:
6:
4:
3:
2:
2205:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2178:Combinatorics
2176:
2175:
2173:
2157:on 2016-01-29
2156:
2152:
2146:
2142:
2141:
2135:
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2098:
2080:
2074:
2066:
2063:
2060:
2056:
2052:
2046:
2042:
2038:
2034:
2029:
2028:
2024:
2016:
2011:
2007:
2003:
2002:
1994:
1991:
1986:
1981:
1977:
1973:
1972:
1964:
1961:
1956:
1951:
1947:
1946:
1938:
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1921:
1917:
1910:
1907:
1903:
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1753:
1749:
1745:
1741:
1740:Combinatorica
1734:
1732:
1728:
1724:
1719:
1716:
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1707:
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1634:
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1587:
1560:
1540:
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1528:
1524:
1520:
1519:A. E. Brouwer
1516:
1497:
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1491:
1485:
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1284:
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1248:
1244:
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1234:
1230:
1226:
1225:
1223:
1216:
1208:
1206:
1192:
1172:
1152:
1132:
1112:
1089:
1086:
1083:
1060:
1052:
1047:
1033:
1013:
993:
985:
981:
980:vertex covers
977:
972:
958:
938:
918:
898:
878:
858:
835:
832:
829:
823:
820:
808:
806:
792:
789:
780:
776:
772:
764:
762:
759:
757:
753:
749:
745:
741:
739:
734:
732:
728:
712:
709:
704:
699:
691:
686:
683:
681:
665:
662:
657:
652:
649:
642:has at least
641:
636:
633:
631:
627:
623:
604:
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589:
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581:
573:
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349:
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317:
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196:
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188:
184:
180:
176:
173:- 1) points,
172:
168:
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155:
153:
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145:
141:
137:
133:
128:
126:
122:
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86:
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78:
74:
70:
66:
62:
58:
50:
48:
46:
42:
38:
34:
30:
26:
22:
2159:, retrieved
2155:the original
2139:
2129:
2109:
2032:
2005:
2004:, Series A,
1999:
1993:
1975:
1974:, Series A,
1969:
1963:
1943:
1937:
1919:
1915:
1909:
1897:
1881:
1877:
1871:
1855:
1851:
1845:
1819:
1813:
1807:
1795:
1783:
1771:
1759:
1743:
1739:
1718:
1706:
1694:
1532:
1531:
1523:A. Schrijver
1399:
1390:László Rédei
1385:
1381:
1377:
1373:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1288:
1278:
1274:
1270:
1266:
1258:
1254:
1250:
1246:
1242:
1238:
1237:
1228:
1227:is a set of
1221:
1219:
1212:
1051:two-coloring
1048:
1006:is a subset
976:hitting sets
973:
931:is a subset
812:
778:
768:
760:
755:
751:
747:
743:
742:
737:
735:
726:
692:has at most
689:
687:
684:
679:
639:
637:
634:
621:
619:
536:is bounded:
533:
529:
525:
519:
467:
457:
453:
449:
445:
441:
440:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
394:
390:
386:
370:
366:
362:
358:
354:
353:
347:
343:
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335:
331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
279:
275:
263:
259:
255:
251:
247:
246:
241:
237:
233:
229:
225:
224:- 2 points,
221:
217:
215:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
156:
151:
147:
139:
135:
131:
129:
124:
120:
116:
112:
106:
96:
93:
84:
80:
76:
72:
68:
64:
60:
55:In a finite
54:
40:
36:
29:blocking set
28:
18:
2183:Hypergraphs
1800:Holder 2001
1746:: 111–114,
1515:Oberwolfach
1273:, of size
1165:. Now both
984:transversal
771:game theory
278:= (1,0,0),
59:π of order
2172:Categories
2161:2016-01-23
2025:References
1378:Rédei line
1145:or within
488:is called
444:: In PG(2,
357:: In PG(2,
250:: In PG(2,
197:and point
89:Fano plane
51:Definition
45:hypergraph
2075:τ
2059:1439-7382
1664:−
1646:−
1638:−
1495:−
1348:≥
1261:+ 2. The
1233:collinear
1220:complete
805:in 1969.
790:≤
595:−
582:≤
566:≤
524:of order
496:τ
213:is in β.
144:collinear
111:of order
85:committee
2106:(1979),
2065:C. Berge
1701:, p. 366
1281:- 1)/2.
758:+ 1)/2.
630:subplane
460:+ 1)/2.
448:), with
373:+ 2)/2.
310:) where
266:+ 1)/2.
201:on line
193:on line
165:in PG(2,
103:Examples
21:geometry
1838:2032754
1802:, p. 45
1253:) with
1239:Theorem
765:History
744:Theorem
528:, PG(2,
520:In the
442:Theorem
361:) with
355:Theorem
254:) with
248:Theorem
107:In any
97:trivial
77:minimal
2147:
2118:
2057:
2047:
1836:
1553:be an
1325:. The
1297:(with
1241:: Let
978:" or "
731:unital
626:square
270:Using
163:side m
1834:JSTOR
1687:Notes
1257:<
1213:In a
1053:" of
736:When
624:is a
620:When
407:X = Y
161:β of
2145:ISBN
2116:ISBN
2055:ISSN
2045:ISBN
1533:Let
1263:dual
1224:-arc
1185:and
813:Let
464:Size
433:and
342:and
318:and
209:and
185:and
27:, a
2037:doi
2010:doi
1980:doi
1950:doi
1924:doi
1886:doi
1860:doi
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1748:doi
1309:in
1301:= |
1105:of
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2096:.)
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2006:24
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1744:14
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517:.
429:,
421:+
417:=
338:,
332:bc
330:=
314:,
304:AC
296:BC
288:AB
242:PQ
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211:AC
207:PQ
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195:AB
181:,
157:A
154:.
91:.
2084:)
2081:H
2078:(
2039::
2012::
1982::
1952::
1926::
1920:3
1888::
1862::
1826::
1820:7
1750::
1670:)
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1093:}
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1084:C
1081:{
1061:H
1034:X
1014:T
994:H
959:X
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836:E
833:,
830:X
827:(
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793:9
756:p
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713:1
710:+
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640:n
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600:q
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570:|
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499:(
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391:X
387:C
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183:B
179:A
175:m
171:m
167:q
152:n
148:n
140:P
136:P
132:n
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117:n
113:n
81:B
73:B
69:B
65:B
61:n
41:m
37:n
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