  Our main result is a new upper bound for the size of k-uniform,
L-intersecting families of sets, where L contains only positive integers. We
characterize extremal families in this setting. Our proof is based on the
Ray-Chaudhuri--Wilson Theorem. As an application, we give a new proof for the
Erd\H{o}s-Ko-Rado Theorem, improve Fisher's inequality in the uniform case and
give an uniform version of the Frankl-F\"uredi conjecture .
