A Proof of a Frankl–Kupavskii Conjecture on Intersecting Families

Abstract

AbstractA family $$\mathcal {F} \subset \mathcal {P}(n)$$ F ⊂ P ( n ) is r-wisek-intersecting if $$|A_1 \cap \dots \cap A_r| \ge k$$ | A 1 ∩ ⋯ ∩ A r | ≥ k for any $$A_1, \dots , A_r \in \mathcal {F}$$ A 1 , ⋯ , A r ∈ F . It is easily seen that if $$\mathcal {F}$$ F is r-wise k-intersecting for $$r \ge 2$$ r ≥ 2 , $$k \ge 1$$ k ≥ 1 then $$|\mathcal {F}| \le 2^{n-1}$$ | F | ≤ 2 n - 1 . The problem of determining the maximum size of a family $$\mathcal {F}$$ F that is both $$r_1$$ r 1 -wise $$k_1$$ k 1 -intersecting and $$r_2$$ r 2 -wise $$k_2$$ k 2 -intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255–1266, 2019). They proved the surprising result that, for $$(r_1,k_1) = (3,1)$$ ( r 1 , k 1 ) = ( 3 , 1 ) and $$(r_2,k_2) = (2,32)$$ ( r 2 , k 2 ) = ( 2 , 32 ) then this maximum is at most $$2^{n-2}$$ 2 n - 2 , and conjectured the same holds if $$k_2$$ k 2 is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for $$(r_1,k_1) = (3,1)$$ ( r 1 , k 1 ) = ( 3 , 1 ) and $$(r_2,k_2) = (2,3)$$ ( r 2 , k 2 ) = ( 2 , 3 ) for all n.