A Sharp Bound for the Number of Sets that Pairwise Intersect at k Positive Values

In this paper we prove that if \( {\user1{L}} \) is a set of k positive integers and {A 1, ..., A m } is a family of subsets of an n-element set satisfying \( {\left| {A_{i} \cap A_{j} } \right|} \in {\user1{L}} \), for all 1 ≤ i < jm, then \( m \leqslant {\sum\nolimits_{i = 0}^k {{\left( {^{{n - 1}}_{i} } \right)}} } \). The case k = 1 was proven 50 years ago by Majumdar.

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Correspondence to Hunter S. Snevily.

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Snevily, H.S. A Sharp Bound for the Number of Sets that Pairwise Intersect at k Positive Values. Combinatorica 23, 527–533 (2003). https://doi.org/10.1007/s00493-003-0031-2

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AMS Subject Classification (2000):

  • 05D05