Cross-intersecting families and primitivity of symmetric systems

  • Authors:

  • Jun Wang;Huajun Zhang

  • Affiliations:

  • Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China;Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China and Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

Let X be a finite set and p@?2^X, the power set of X, satisfying three conditions: (a) p is an ideal in 2^X, that is, if A@?p and B@?A, then B@?p; (b) for A@?2^X with |A|=2, A@?p if {x,y}@?p for any x,y@?A with xy; (c) {x}@?p for every x@?X. The pair (X,p) is called a symmetric system if there is a group @C transitively acting on X and preserving the ideal p. A family {A"1,A"2,...,A"m}@?2^X is said to be a cross-p-family of X if {a,b}@?p for any a@?A"i and b@?A"j with ij. We prove that if (X,p) is a symmetric system and {A"1,A"2,...,A"m}@?2^X is a cross-p-family of X, then@?i=1m|A"i|==|X|@a(X,p), where @a(X,p)=max{|A|:A@?p}. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-t-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.