Restricted -wise -intersecting families on set systems

Let L = { λ 1 , … , λ s } be a set of s non-negative integers with λ 1 < λ 2 < ⋯ < λ s , and let t ≥ 2 . A family F of subsets of an n -element set is called t -wise L -intersecting if the cardinality of the intersection of any t distinct members in F belongs to L . We give the following improvement to the Füredi–Sudakov theorem. Let t ≥ 3 and F be a t -wise L -intersecting family of subsets of [ n ] . Then, for | ⋂ F ∈ F F | < λ 1 , | F | = o ( n s ) ; for | ⋂ F ∈ F F | ≥ λ 1 , and n sufficiently large, | F | ≤ k + s − 1 s + 1 n − λ 1 s + ∑ i ≤ s − 1 n − λ 1 i . We also give a sharp upper bound for the size of a k -uniform t -wise L -intersecting family when s = 1 .