Fractional cross intersecting families

2019/03/05 by Mathew, Rogers, Ray, Ritabrata, Srivastava, Shashank
#05D05 #68R05 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.01872

Abstract

Let A=\A₁,...,A_p\ and B=\B₁,...,B_q\ be two families of subsets of [n] such that for every i∈ [p] and j∈ [q], |A_i∩ B_j|= (c)/(d)|B_j|, where (c)/(d)∈ [0,1] is an irreducible fraction. We call such families "(c)/(d)-cross intersecting families". In this paper, we find a tight upper bound for the product |A||B| and characterize the cases when this bound is achieved for (c)/(d)=(1)/(2). Also, we find a tight upper bound on |A||B| when B is k-uniform and characterize, for all (c)/(d), the cases when this bound is achieved.

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• Maximal fractional cross-intersecting families
• r-wise fractional L-intersecting family
• The maximum product of sizes of cross-t-intersecting uniform families
• r-cross t-intersecting families via necessary intersection points