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.