Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any a, 7r H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible...Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any a, 7r H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible imprimitive reflection group G（m,p,n） is a coset of the stabilizer of a point in {1,... ,n} provided n is sufficiently large.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
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Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any a, 7r H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible imprimitive reflection group G（m,p,n） is a coset of the stabilizer of a point in {1,... ,n} provided n is sufficiently large.
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}.
