A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we d...A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.展开更多
For a graph G and an integer r ≥ 1, G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star （a family of independent r-sets containing some fixed vertex in G）, and G is stric...For a graph G and an integer r ≥ 1, G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star （a family of independent r-sets containing some fixed vertex in G）, and G is strictly r-EKR if every extremal intersecting family of independent r-sets is a star. Recently, Hurlbert and Kamat gave a preliminary result about EKR property of ladder graphs. They showed that a ladder graph with n rungs is 3-EKR for all n ≥3. The present paper proves that this graph is r-EKR for all 1 ≤ r 〈 n, and strictly r-EKR except for r = n - 1.展开更多
For k = （k1,...,kn） E Nn, 1 ≤ k1 ≤ ... ≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr：= {{（a1,la1）,...,（ar,lar）} ： {a1,...,ar} [n],lai ∈ [kai],i = 1,...,r}. A family A of labeled r-sets i...For k = （k1,...,kn） E Nn, 1 ≤ k1 ≤ ... ≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr：= {{（a1,la1）,...,（ar,lar）} ： {a1,...,ar} [n],lai ∈ [kai],i = 1,...,r}. A family A of labeled r-sets is intersecting if any two sets in ~4 intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.展开更多
让是一个有限集合，并且让 G 是一个排列组在上。如果， G 的子集 H 被称为交叉为任何东西， H，他们同意至少一个点。我们证明无法缩减的 imprimitive 思考的一个最大的交叉子集组织 G (m， p， n ) 是一个点在的 stabilizer 的 coset ...让是一个有限集合，并且让 G 是一个排列组在上。如果， G 的子集 H 被称为交叉为任何东西， H，他们同意至少一个点。我们证明无法缩减的 imprimitive 思考的一个最大的交叉子集组织 G (m， p， n ) 是一个点在的 stabilizer 的 coset { 1，， n } 提供的 n 足够地大。展开更多
基金Supported by the doctoral Foundation of Yanshan University(No.B314)
文摘A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.
基金Supported by the National Natural Science Foundation of China(No.11201409,No.11371327)the Natural Science Foundation of Hebei Province of China(No.A2013203009)
文摘For a graph G and an integer r ≥ 1, G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star （a family of independent r-sets containing some fixed vertex in G）, and G is strictly r-EKR if every extremal intersecting family of independent r-sets is a star. Recently, Hurlbert and Kamat gave a preliminary result about EKR property of ladder graphs. They showed that a ladder graph with n rungs is 3-EKR for all n ≥3. The present paper proves that this graph is r-EKR for all 1 ≤ r 〈 n, and strictly r-EKR except for r = n - 1.
基金Supported by the National Natural Science Foundation of China (No. 11001249)the Mathematical Tianyuan Foundation of China (No. 11026180)
文摘For k = （k1,...,kn） E Nn, 1 ≤ k1 ≤ ... ≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr：= {{（a1,la1）,...,（ar,lar）} ： {a1,...,ar} [n],lai ∈ [kai],i = 1,...,r}. A family A of labeled r-sets is intersecting if any two sets in ~4 intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.
基金Acknowledgements The author would like to express her deep gratitude to Professor Jun Wang for guiding her into this area and thank the referees for their invaluable suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11001176, 10971138）.
文摘让是一个有限集合，并且让 G 是一个排列组在上。如果， G 的子集 H 被称为交叉为任何东西， H，他们同意至少一个点。我们证明无法缩减的 imprimitive 思考的一个最大的交叉子集组织 G (m， p， n ) 是一个点在的 stabilizer 的 coset { 1，， n } 提供的 n 足够地大。
