偏序集的Erd（？）s-Ko-Rado性质及相关问题
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摘要
论文主要研究偏序集的Erd(?)s-Ko-Rado(EKR)性质,交族的正规匹配(NM)性质、LYM性质,子空间族的最小下影问题以及Erd(?)s-Ko-Rado定理在置换群中的模拟。
本文分四章。第一童介绍EKR定理的内容、发展历史以及几种经典证明方法。第二章研究分次偏序集中交族的正规匹配(NM)性质和LYM性质。在较广的意义下我们给出Sperner型性质和交性质的一个统一的处理,于是就自然地引入了分次偏序集中交族的NM性质和LYM性质,证明交-NM性质蕴含交-LYM性质(反之不然)。接着用移位算子证明了B_n的严格交-NM性质。第三章研究子空间族的最小下影问题。用群作用的方法得到L_n(q)到B_n的保秩且保序的映射,这样就可以把L_n(q)看作是加权的B_n。由此得到一个相应的Kruskal-Katona定理,它可以看作是加权的Kruskal-Katona定理。第四章研究对称群和Coxetcr群的EKR性质,通过计算不动点,我们给出对称群的严格EKR性质的一个简单证明,在此基础上证明了Coxeter群的严格EKR性质。
This thesis investigates the Erd(o|¨)s-Ko-Rado(EKR) property of intersecting families, the normalized matching(NM) property and the LYM property of intersecting antichains. the shadow minimization problem of subspace families,and analogs of Erd(o|¨)s-Ko-Rado Theorem in permutation groups.
The thesis consists of four chapters.The first contributes to the contents,the history, and the classic proofs of Erd(o|¨)s-Ko-Rado Theorem.The second devotes to the NM property and the LYM property of intersecting antichains.In a more general setting we give a unified treatment of the Sperner-type properties and the intersecting property in a ranked poset.Then,the LYM property and the(strict) normalized matching(NM) property for intersecting families in a ranked poser are introduced in a natural way.We prove that the NM property implies the LYM property for intersecting families,but the converse is not generally true.By the shift operation we establish the strict NM property for intersecting families in the boolean lattices.The third investigates the shadow minimization problem of subspace families.By the group action method,we obtain a rank-preserving and order-preserving map from L_n(q) to B_n,so that L_n(q) can be considered as weighted B_n,thus we yield a corresponding Kruskal-Katona Theorem.which can be considered as a weighted Kruskal-Katona Theorem.The fourth considers the EKR property of symmetric groups and Coxeter groups.By analyzing the fixed points of permutations,we present a short proof of EKR property of symmetric groups.Based on this we establish EKR-type theorems for Coxeter groups of types B and D.
本文分四章。第一童介绍EKR定理的内容、发展历史以及几种经典证明方法。第二章研究分次偏序集中交族的正规匹配(NM)性质和LYM性质。在较广的意义下我们给出Sperner型性质和交性质的一个统一的处理,于是就自然地引入了分次偏序集中交族的NM性质和LYM性质,证明交-NM性质蕴含交-LYM性质(反之不然)。接着用移位算子证明了B_n的严格交-NM性质。第三章研究子空间族的最小下影问题。用群作用的方法得到L_n(q)到B_n的保秩且保序的映射,这样就可以把L_n(q)看作是加权的B_n。由此得到一个相应的Kruskal-Katona定理,它可以看作是加权的Kruskal-Katona定理。第四章研究对称群和Coxetcr群的EKR性质,通过计算不动点,我们给出对称群的严格EKR性质的一个简单证明,在此基础上证明了Coxeter群的严格EKR性质。
This thesis investigates the Erd(o|¨)s-Ko-Rado(EKR) property of intersecting families, the normalized matching(NM) property and the LYM property of intersecting antichains. the shadow minimization problem of subspace families,and analogs of Erd(o|¨)s-Ko-Rado Theorem in permutation groups.
The thesis consists of four chapters.The first contributes to the contents,the history, and the classic proofs of Erd(o|¨)s-Ko-Rado Theorem.The second devotes to the NM property and the LYM property of intersecting antichains.In a more general setting we give a unified treatment of the Sperner-type properties and the intersecting property in a ranked poset.Then,the LYM property and the(strict) normalized matching(NM) property for intersecting families in a ranked poser are introduced in a natural way.We prove that the NM property implies the LYM property for intersecting families,but the converse is not generally true.By the shift operation we establish the strict NM property for intersecting families in the boolean lattices.The third investigates the shadow minimization problem of subspace families.By the group action method,we obtain a rank-preserving and order-preserving map from L_n(q) to B_n,so that L_n(q) can be considered as weighted B_n,thus we yield a corresponding Kruskal-Katona Theorem.which can be considered as a weighted Kruskal-Katona Theorem.The fourth considers the EKR property of symmetric groups and Coxeter groups.By analyzing the fixed points of permutations,we present a short proof of EKR property of symmetric groups.Based on this we establish EKR-type theorems for Coxeter groups of types B and D.
引文
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[3]R.Ahlswede,H.Aydinian and L.H.Khachatrian,The intersection theorem for direct products,European J.Combin.,19(1998),649-661.
[4]I.Anderson.Cornbinatorics of Finite sets,Oxford University Press,Oxford,1987.
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[6]C.Bey,The Erd(o|¨)s-Ko-Rado bound for the function lattice,Discrete Appl.Math.,95(1999),115-125.
[7]C.Bey,An intersection theorem for weighted sets,Discrete Math.,235(2001),145-150.
[8]S.Bezrukov and A.Blokhuis,A Kruskal-Katona Type Theorem for The Linear Lattice,Europ.J.Combinatorics,20(1999),123-130.
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[10]A.BjSrner and F.Brcnti,Combinatorics of Coxeter groups,in:Graduate Texts in Mathernatics,vol.231,Springer,New York,2005.
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[13]M.Bóna,Combinatorics of permutations,Discrete Mathematics and its Applications(Boca Raton).Chapman & Hall/CRC,Boca Raton,FL,2004.
[14]P.J.Cameron and C.Y.Ku,Intersecting families of permutations,European J.Combin.,24(2003),881-890.
[15]P.J.Cameron,Permutations,in G.Halász,L.Lovász,M.Simonovits and V.T.Sós,editors,Paul Erd(o|¨)s and his Mathematics,Vol.Ⅱ,205-239,Bolyai Society Mathematical Studies 11,Springer,Berlin,2002.
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[17]L.Comtet,Advanced Combinatorics,Rcidel,Dordrecht,1974.
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[19]E.Czabarka,Shifting in finite vector spaces,Ph.D Thesis,University of South Carolina,1998,UMI Number:9918919.
[20]D.E.Daykin,ErdSs-Ko-Rado from Kruskal-Katona,J.Combinatorial Theory Ser.A,20(1974),254-255.
[21]M.Deza and P.Frankl,Erd(o|¨)s-Ko-Rado theorem-22 years later,SIAM J.Algebraic Discrete Methods,4(1983),419-431.
[22]M.Deza and P.Frankl,On the maximum number of permutations with given maximal or minimal distance,J.Combin.Theory Set.A,22(1977),352-362.
[23]R.P.Dilworth,A decomposition theorem for partially ordered sets,Ann.Math.,51(1950),161-165.
[24]K.Engel,Strong properties in partially ordered sets Ⅱ,Discrete Math.,48(1984),187-196.
[25]K.Engel,An ErdSs-Ko-Rado theorem for the subcubcs of a cube,Combinatorica,4(1984),133-140.
[26]K.Engel,Sperner Theory,Cambridge University Press,Cambridge,1997.
[27]P.Erd(o|¨)s,On a lemma of Littlewood and Offord,Bull.Amer.Math.Soc.,51(1945),898-902.
[28]P.Erd(o|¨)s,My joint work with Richard Rado,in C.Whitehead,editor,Surveys in combinatorics 1987,London Math.Soc.Lecture Note Ser.,123,53-80,Cambridge Univ.Press,Cambridge,1987.
[29]P.L.Erd(o|¨)s,U.Faigle and W.Kern,A group-theoretic setting for some intersecting Sperner families,Combin.Probab.Comput.,1(1992),323-334.
[30]P.Erd(o|¨)s and D.J.Kleitman,Extremal problems among subsets of a set,Discrete Math.,306(2006),923-931.
[31]P.Erd(o|¨)s,C.Ko and R.Rado,Intersection theorems for systems of finite sets,Quart.J.Math.Oxford Ser.,2(1961),313-318.
[32]P.Frankl,The Erd(o|¨)s-Ko-Rado theorem is true for n=ckt,in Combinatorics(Proc.Fifth Hungarian Colloq.,Keszthely,1976),Vol.Ⅰ,365-375,North-Holland,Amsterdam,1978.
[33]P.Frankl,A new short proof for the Kruskal-Katona theorem,Discrete Math.,48,327-329.
[34]P.Frankl and Z.Furedi,The Erd(o|¨)s-Ko-Rado theorem for integer sequences,SIAM J.Algebraic Discrete Math.,1(1980),376-381.
[35]P.Frankl and N.Tokushige,Thc Erd(o|¨)s-Ko-Rado theorcm for integer sequences,Combinatorica,19(1999),55-63.
[36]P.Frankl and R.M.Wilson.The Erd(o|¨)s-Ko-Rado theorem for vector spaces,J.Combin,Theory Ser.A,43(1986),no.2,228-236.
[37]R.Freese,An application of Dilworth's lattice of maximal antichains,Discrete Math.,7(1974),107-109.
[38]C.Godsil and K.Meagher,A new proof of the Erd(o|¨)s-Ko-Rado theorem for intersecting families of permutations,arXiv:0710.2109v1[math.COl 10 Otc 2007.
[39]C.Greene,G.Katona and D.J.Kleitman,Extensions of the Erd(o|¨)s-Ko-Rado theorem,Studies in Appl.Math.,55(1976),1-8.
[40]C.Greene and D.J.Kleitman,The structure of Sperner k-family,J.Combin.Theory Ser.A,20(1976),41-68.
[41]C.Greene and D.J.Kleitman,Proof techniques in the theory of finite sets,In G.-C.Rota,editor,Studies in Combinatorics,MAA Studies in Math.,Vol.17,22-79,Math.Assoc.America,Washington DC,1978.
[42]J.R.Griggs.Sufficient conditions for a symmetric chain order,SIAM J.Appl.Math.,32(1977),807-809.
[43]J.R.Griggs,On Chains and Sperncr k-families in ranked posets,J.Combin.Theory Ser.A,28(1980),156-168.
[44]A.J.W.Hilton and E.C.Milner,Some intersection theorems for systems of finite sets.Quart.J.Math.Oxford,18(1967),369-384.
[45]F.Holroyd and J.Talbot,Graphs with the Erd(o|¨)s-Ko-Rado property,Discrete Math.,293(2005),165-176.
[46]R.Howard,G.Karolyi and L.A.Szekely,Towards a Katona type proof for the 2-intersecting Erd(o|¨)s-Ko-Rado theorem,Elec.J.Combin.,8(2001),#R31.
[47]W.N.Hsieh,Intersection theorems for systems of finite vector spaces,Discrete Math.,12(1975),1-16.
[48]W.N.Hsich and D.J.Kleitman,Normalized matching in direct products of partial orders,Studies in Appl.Math.,52(1973),285-289.
[49]G.O.H.Katona,A theorem on finite sets,Theory of graphs,Proc.Colloq.Tihany,1966,187-207,Academic Press,New York,1968.
[50]G.O.H.Katona,A simple proof of the Erd(o|¨)s-Chao Ko-Rado theorem,J.Combin Theory Ser.B,13(1972),183-184.
[51]G.O.H.Katona,A simple proof of a theorem of Milner,J.Combin.Theory Set.A,83(1998),138-140.
[52]G.O.H.Katona,Extremal problems for finite scts and convex hulls-a survey,Discrete Math.,164(1997),175-185.
[53]D.Kleitman,M.Edelberg and D.Lubell,Maximal-sized antichains in partial orders,Discrete Math.,1(1971),47-53.
[54]D.Kleitman,On an extremal property of antichains in partial orders,the LYM property and some of its implications and application,In M.Hall and J.H.van Lint,editors,Combinatorics,Part 2:Graph theory,foundations,partitions and combinatorial geometry,77-90,Math.Centrum,Amsterdam,1974.
[55]D.E.Knuth,Subsets,subspaces,and partitions,J.Combin.Theory,10(1971),178-180.
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