On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let \( {\user1{A}} \) be a family of k-subsets of ℕ. The ℓ-shadow of \( {\user1{A}} \) is the set of all (k-ℓ)-subsets \( \partial _{{\ell }} {\user1{A}} \) contained in the members of \( {\user1{A}} \). Let \( {\user1{A}} \) be a t-intersecting family (any two members have at least t elements in common) with \( {\left| {\user1{A}} \right|} = m \). Given k,t,m the problem is to minimize \( {\left| {\partial _{{\ell }} {\user1{A}}} \right|} \) (over all choices of \( {\user1{A}} \)). In this paper we solve this problem when m is big enough.

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Correspondence to R. Ahlswede.

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Ahlswede, R., Aydinian, H. & Khachatrian, L.H. On Shadows Of Intersecting Families. Combinatorica 24, 555–566 (2004). https://doi.org/10.1007/s00493-004-0034-7

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Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05