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1、组合设计的大集与超大集已解决的和待解决的康庆德河北师范大学数学研究所2009.7.29Kirkman’sschoolgirlproblem(T.P.Kirkman1847)SUNMONTUEWEDTHUFRISAT大集问题的起源和背景ThomasPenyngtonKirkman(英格兰教会的教区长)*2{a,50,31},{01,41,51},{00,10,11},{20,40,61},{30,60,21}SUNMONTUEWEDTHUFRISAT(1850Sylvester,Cayley1974Denniston)3*经典三元系的大集与超
2、大集LSTS,LMTS,LDTS,LHTS,OLSTS,OLMTS,OLDTS.其它三元系的大集与超大集LT1,LT2,LT3,OLT1,OLT2,OLT3;LESTS,LEMTS,LEDTS.纯的有向三元系的大集与超大集LPMTS,LPDTS,OLPMTS,OLPDTS.可分解（几乎可分解）三元系的大集与超大集LKTS,LRMTS,LRDTS,OLKTS,OLRMTS,OLRDTS.LARMTS,LARDTS,OLARMTS,OLARDTS.图设计的大集与超大集路分解：P3-LGD,OP3-LGD,P3-OLGD,OP3-OLGD,P4-LGD,Pk-LGD.星(圈)分解：K1,3-LGD,
3、K1,4-LGD,K1,k-LGD;C4-LGD.Hamilton圈(路)分解：LHCD,LHPD,LDHCD,LDHPD;LCS(v,v-1,λ).可分组设计的大集LGDD.拉丁方的大集LDILS,Golfdesign,...t-设计的大集LSλ(t,k,v)…11:12:054基本文献C.J.Colbourn&J.H.Dinitz,TheCRCHandbookofCombinatorialDesigns,CRCPress(SecondEdition),2006.J.H.Dinitz&D.R.Stinson,ContemporaryDesignTheory–Acollectionofsurv
4、eys,Wiley,1992.Q.D.Kang,Onlargesetsofcombinatorialdesigns,AdvanceofMathematics,32(2003),269-284.*5A.经典三元系的大集与超大集LSTS,LMTS,LDTS,LHTS, OLSTS,OLMTS,OLDTS,OLHTS, LPMTS,LPDTS,OLPMTS,OLPDTS.*6Sixtypesoftriplesand thecorrespondingtriplesystems*7Theexistenceoftriplesystems*8*9经典三元系大集的存在谱*AshortproofforLSTS(
5、v)wasgivenbyL.Ji.*Lindner,Street,Colbourn,RosaandTeirlinckalsogavesomeresultsforLMTS(v).11:12:0510经典三元系超大集的存在谱*遗留问题：11:12:0511LargeSetsofpureorintedtriplesystems*遗留问题:*12OverlargeSetsofpureorintedtriplesystems*遗留问题:*13B.其它三元系的大集与超大集LT1,LT2,LT3,OLT1,OLT2,OLT3,LESTS,LEMTS,LEDTS.*14mixedtriplesystems—T
6、1,T2,T3*15*16*17ConclusionsforLTiandOLTi*遗留问题:Q.Kang,Z.Tian&L.Yuan,2003-200711:12:0518Aclassicaltripleconsistsofthreedistinctelements,butanextendedtripleisallowedtocontainrepeatedelements.STS,MTS,DTS(LSTS,LMTS,LDTS)⇒ESTS,EMTS,EDTS(LESTS,LEMTS,LEDTS).Extendedtriplesystems*19ExamplesofLESTS*20Examples
7、ofLEMTS*21ExamplesofLEDTS*22AconstructionforLEDTS(7)*23ThereexistLEDTS(v)forThereexistLESTS(v)andLEMTS(v)*24C.可分解三元系的大集与超大集LKTS,LRMTS,LRDTS, OLKTS,OLRMTS,OLRDTS, LARMTS,LARDTS, OLARMTS,OLARDTS.*25LKTS
