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Multiply Intersecting Families.

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  • UserAgnijo Banerjee (Cambridge)
  • ClockThursday 22 May 2025, 14:30-15:30
  • HouseMR12.

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A family F āŠ‚ P(n) is r-wise k-intersecting if |A1 āˆ© Ā· Ā· Ā· āˆ© Ar| ā‰„ k for any A1, . . . , Ar āˆˆ F. It is easily seen that if F is r-wise k-intersecting for r ā‰„ 2, k ā‰„ 1 then |F| ā‰¤ 2 . The problem of determining the maximal size of a family F that is both r1-wise k1-intersecting and r2-wise k2-intersecting was raised in 2019 by Frankl and Kupavskii. They proved the surprising result that, for (r1, k1) = (3, 1) and (r2, k2) = (2, 32) then this maximum is at most 2(nāˆ’2) , and conjectured the same holds if k2 is replaced by 3. In this talk I shall not only prove this conjecture but also determine the exact maximum for (r1, k1) = (3, 1) and (r2, k2) = (2, 3) for all n.

This talk is part of the Combinatorics Seminar series.

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