Set systems with restricted kk-wise LL-intersections modulo a prime number

The classical Erdös–Ko–Rado theorem on the size of an intersecting family of tt-subsets of the set {1,2,…,n}{1,2,…,n} is one of the most basic intersection theorems for set systems. Since the Erdös–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 kk-wise LL-intersecting and cross-intersecting families by employing the existing linear algebra methods.