Set systems with restricted k-wise L-intersections modulo a prime number

The classical Erdos-Ko-Rado theorem on the size of an intersecting family of t-subsets of the set {1,2,...,n} is one of the most basic intersection theorems for set systems. Since the Erdos-Ko-Rado theorem was published, there have been many intersection theorems on set systems appeared in the literature, such as the well-known Frankl-Wilson theorem, Alon-Babai-Suzuki theorem, Grolmusz-Sudakov theorem, and Qian-Ray-Chaudhuri theorem. In this paper, we will survey results on intersecting families and derive extensions for these well-known intersection theorems to k-wise L-intersecting and cross-intersecting families by employing the existing linear algebra methods.