Restricted tt-wise LL-intersecting families on set systems

Let L={λ1,…,λs}L={λ1,…,λs} be a set of ss non-negative integers with λ1<λ2<⋯<λsλ1<λ2<⋯<λs, and let t≥2t≥2. A family FF of subsets of an nn-element set is called tt-wise LL-intersecting if the cardinality of the intersection of any tt distinct members in FF belongs to LL. We give the following improvement to the Füredi–Sudakov theorem. Let t≥3t≥3 and FF be a tt-wise LL-intersecting family of subsets of [n][n]. Then, for |⋂F∈FF|<λ1|⋂F∈FF|<λ1, |F|=o(ns);|F|=o(ns); for |⋂F∈FF|≥λ1|⋂F∈FF|≥λ1, and nn sufficiently large, |F|≤k+s−1s+1n−λ1s+∑i≤s−1n−λ1i. We also give a sharp upper bound for the size of a kk-uniform tt-wise LL-intersecting family when s=1s=1.