A k-uniform family of sets F is d-wise, t-intersecting if the intersection of any d sets from F contains at least t elements. In this talk, I will state and prove analogues of the classical Erdos-Ko-Rado and Hilton-Milner theorems for d-wise, t-intersecting families, improving upon several earlier results by a number of authors, including O'Neill-Verstraete and Tokushige. I will also discuss some possible directions for future research. Time permitting, I will indicate how d-wise, t-intersecting families can be used to construct K_{s,t}-intersecting families of size larger than the trivial construction, answering a question of Ellis. This is joint work with Jozsef Balogh.
