Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem Huajun Zhang Department of Mathematics, Zhejiang Normal University Zhejiang 321004, P. R. China December 27, 2013, Shanghai Jiao Tong University Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (EKR Theorem) If A is an intersecting family of k-subsets of [ n ] = { 1 , 2 , . . . , n } , i.e., A ∩ B � = ∅ for any A , B ∈ A , then � n − 1 � |A| ≤ k − 1 subject to n ≥ 2 k. Equality holds if and only if every subset in A contains a common element of [ n ] except for n = 2 k. P. Erd˝ os, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser., 2 (1961), 313-318. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (EKR Theorem for Finite Vector Spaces) If A is an intersecting family of k-dimensional subspaces of an n-dimensional vector space over the q-element field, i.e., dim( A ∩ B ) ≥ 1 for any A , B ∈ A , then � n − 1 � |A| ≤ k − 1 subject to n ≥ 2 k. Equality holds if and only if every subset in A contains a common nonzero vector except the case n = 2 k. W. N. Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math., 12 (1975), 1-16. C. Greene and D. J. Kleitman, Proof techniques in the ordered sets, in: G.-C. Rota, ed., “Studies in Combinatorics” 1978, 22-79. P. Frankl, R. M. Wilson, The Erd˝ os-Ko-Rado theorem for vector spaces, JCTA 43 (1986), 228-236. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (EKR Theorem for Permutations) If A is an intersecting family in S n (the symmetric group on [ n ] ), i.e., for each pair σ, τ ∈ S n there is an i ∈ [ n ] with σ ( i ) = τ ( i ) , then |A| ≤ ( n − 1)! . Equality holds if and only if A is a coset of the stabilizer of a point. M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, JCTA 22(1977) 352-362. P. Cameron and C.Y. Ku, Intersecting families of permutations, EuJC 24 (2003), 881-890. JW, J. Zhang, An Erd˝ os-Ko-Rado-type theorem in Coxeter groups, EuJC 29 (2008), 1112-1115. D. Ellis, E. Friedgut, H. Pilpel, Intersecting families of permutations, Journal of the American Mathematical Society 24 (2011) 649-682. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem A q -signed k -set is a pair ( A , f ), where A ⊆ [ n ] is a k -set and f is a function from A to [ q ]. A family F of q -signed k -sets is intersecting if for any ( A , f ), ( B , g ) ∈ F there exists x ∈ A ∩ B such that f ( x ) = g ( x ). � [ n ] } and B n ( q ) = � n Set B k � i =0 B k n ( q ) = { ( A , f ) : A ∈ n ( q ). k � [ n ] A r -partial permutation of [ n ] is a pair ( A , f ) with A ∈ � r and f is an injective map from A to [ n ]. the set of all r -partial permutations of [ n ] denoted by P r , n . Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (EKR Theorem for Signed Sets) (Bollob´ as and Leader) Fix a positive integer k ≤ n, and let F be an intersecting family of q-signed k-sets on [ n ] , where q ≥ 2 . Then � n − 1 q k − 1 . Unless q = 2 and k = n, equality holds if and � |F| ≤ k − 1 only if F consists of all q-signed k-sets ( A , f ) such that x 0 ∈ A and f ( x 0 ) = ε 0 for some fixed x 0 ∈ [ n ] , ε 0 ∈ [ q ] . Bollobas B. Bollob´ as and I. Leader, An Erd¨ os-Ko-Rado theorem for signed sets, Comput. Math. Applic. 34 (11) (1997) 9-13. Y.S. Li, J. Wang, Erd˝ os-Ko-Rado-Type Theorems for Colored Sets, Electron. J. Combin. 14 (1) (2007). Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (EKR Theorem for Partial Permutation) Fix a positive integer r < n, and let F be an intersecting family of � ( n − 1)! � n − 1 P r , n . Then |F| ≤ ( n − r )! . Equality holds if and only if F r − 1 consists of all r-partial permutations ( A , f ) such that i ∈ A and f ( i ) = j for some fixed i , j ∈ [ n ] . C. Y. Ku and I. Leader, An Erd¨ os-Ko-Rado theorem for partial permutations, Disc. Math. 306 (2006) 74-86. Y.S. Li, J. Wang, Erd˝ os-Ko-Rado-Type Theorems for Colored Sets, Electron. J. Combin. 14 (1) (2007). Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (Hilton, 1977) Let A 1 , A 2 , . . . , A m be cross-intersecting families of k subsets of [ n ] with A 1 � = ∅ , i.e., for any A i ∈ A i and A j ∈ A j , i � = j, A i ∩ A j � = ∅ . If k ≤ n / 2 , then m � n � � , if m ≤ n / k; � k |A i | ≤ � n − 1 � m , if m ≥ n / k. k − 1 i =1 Unless m = 2 = n / k, the bound is attained if and only if one of the following holds: � [ n ] � (i) m ≤ n / k and A 1 = , and A 2 = · · · = A m = ∅ ; k � n − 1 � (ii) m ≥ n / k and |A 1 | = |A 2 | = . . . = |A m | = . k − 1 A.J.W. Hilton, An intersection theorem for a collection of families of subsets of a finite set, J. London Math. Soc. 2 (1977) 369-384. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Results The Hilton Theorem was generalized to partial permutation, signed sets and labled sets. P. Borg, Cross-intersecting families of permutations, J. Combin. Theory Ser. A, 117 (2010) 483-487. P. Borg, Intersecting and cross-independent families of labeled sets, Electron. J. Combin. 15 (2008) N9. P. Borg and I. Leader, Multiple cross-intersecting families of signed sets, J. Combin. Theory Ser. A, 117 (2010) 583-588. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Results The Hilton Theorem was generalized to general case. J. Wang, H.J. Zhang, Cross-intersecting families and primitivity of symmetric systems, J. Combin. Theory Ser. A 118 (2011) 455-462. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (Hilton and Milner 1967) � [ n ] � Let n and a be two positive integers with n ≥ 2 a. If A , B ⊆ a with A ∩ B � = ∅ for all A ∈ A and B ∈ B , then � n � � n − a � |A| + |B| ≤ − + 1 . a a A.J.W. Hilton and E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford 18 (1967) 369-384. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem ( Frankl and Tohushige) Let n, a and b be three positive integers with n ≥ a + b and a ≤ b. � [ n ] � [ n ] � � If A ⊆ and B ⊆ with A ∩ B � = ∅ for all A ∈ A and a b B ∈ B , then � n � � n − a � |A| + |B| ≤ − + 1 . b b P. Frankl and N. Tohushige, Some best possible inequalities concerning cross-intersecting families, J. Combin. Theory Ser. A 61 (1992) 87-97. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Results The Hilton-Milner Theorem was generalized to the general cases. J. Wang, H.J. Zhang, Nontrivial independent sets of bipartite graphs and cross-intersecting families, J. Combin. Theory Ser. A, 120 (2013) 129-141. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem � [ n ] � [ n ] If A ⊆ � and B ⊆ � are cross-intersecting with k , ℓ ≤ n / 2 , k ℓ then � n − 1 �� n − 1 � |A||B| ≤ . k − 1 ℓ − 1 � [ n ] � Moreover, the equality holds if and only if A = { A ∈ : i ∈ A } k � [ n ] � and B = { B ∈ : i ∈ B } for some i ∈ [ n ] , unless n = 2 k = 2 ℓ . ℓ L. Pyber, A new generalization of the Erd˝ os-Rado-Ko theorem, J. Combin. Theory Ser. A, 43 (1986) 85-90. M. Matsumoto, N. Tokushige, The exact bound in the Erd˝ os-Rado-Ko theorem for cross-intersecting families, J. Combin. Theory Ser. A, 52 (1989) 90-97. C. Bey, On cross-intersecting families of sets, Graphs Combin., 21 (2005) 161-168. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (Tokushige) Let p be a real with 0 < p < 0 . 114 , and let t be an integer with 1 ≤ t ≤ 1 / (2 p ) . For fixed p and t there exist positive constants ε , n 1 such that for all integers n , k with n > n 1 and | k n − p | < ε ,the � [ n ] � [ n ] � � following is true: if two families A 1 ⊂ and A 2 ⊂ are k k cross t-intersecting, then � 2 � n − t |A 1 ||A 2 | ≤ k − t � [ n ] � with equality holding iff A 1 = A 2 = { F ∈ : [ t ] ⊂ F } (up to k isomorphism). N. Tokushige, On cross t-intersecting families of sets, J. Combin. Theory Ser. A, 117 (2010)1167-1177. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
1. Erd˝ os-Ko-Rado Theorem Theorem (Ellis, Friedgut, Pilpel) For any positive integer k and any n sufficiently large depending on k, if I , J ⊂ S n are k-cross-intersecting, then | I || J | ≤ (( n − k )!) 2 . Equality holds if and only if I = J and I is a k-coset of S n . D. Ellis, E. Friedgut, H. Pilpel, Intersecting families of permutations, Journal of the American Mathematical Society 24 (2011) 649-682. Huajun Zhang Some Problems and Generalizations on Erd˝ os-Ko-Rado Theorem
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