Let Cnk denote the set of all t -subsets of an n -set. Assume A ∈ Cnk and B∈Cn.b (A,B) is called a cross-2-intersecting family if |A∩B|≥2 for any A∈A, B∈B.In this paper, the best upper bounds of the cardinalitie...Let Cnk denote the set of all t -subsets of an n -set. Assume A ∈ Cnk and B∈Cn.b (A,B) is called a cross-2-intersecting family if |A∩B|≥2 for any A∈A, B∈B.In this paper, the best upper bounds of the cardinalities for non-empty cross-2-intersecting families of a- and i- subsets are obtained for some n and b . A new proof for a Frankl-Tokushige theorem [6] is also given.展开更多
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
基金Suppored by Postdoctral Fellowship Foundation of China
文摘Let Cnk denote the set of all t -subsets of an n -set. Assume A ∈ Cnk and B∈Cn.b (A,B) is called a cross-2-intersecting family if |A∩B|≥2 for any A∈A, B∈B.In this paper, the best upper bounds of the cardinalities for non-empty cross-2-intersecting families of a- and i- subsets are obtained for some n and b . A new proof for a Frankl-Tokushige theorem [6] is also given.
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.
