Restricted t-wise L-intersecting families on set systems

Let L={@l"1,...,@l"s} be a set of s non-negative integers with @l"1 =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 Furedi-Sudakov theorem. Let t>=3 and F be a t-wise L-intersecting family of subsets of [n]. Then, for |@?"F"@?"FF| [email protected]"1, and n sufficiently large, |F|@[email protected]"[email protected][email protected][email protected]"1i. We also give a sharp upper bound for the size of a k-uniform t-wise L-intersecting family when s=1.