Set systems with cross -intersection and -wise -intersecting families

We prove some results involving cross L -intersections of two families of subsets of [ n ] = { 1 , 2 , … , n } . As a consequence, we derive the following results: (1) Let L = { l 1 , l 2 , … , l s } be a set of s positive integers. If F = { F 1 , F 2 , … , F m } is a family of subsets of X = [ n ] satisfying | F i − F j | ∈ L for i ≠ j , then m ≤ ∑ i = 0 s n − 1 i . (2) Let p be a prime, k ≥ 2 , and L = { l 1 , l 2 , … , l s } and K = { k 1 , k 2 , … , k r } be two disjoint subsets of { 0 , 1 , … , p − 1 } . Suppose F is a family of subsets of [ n ] such that | F i | ( mod p ) ∈ K for all F i ∈ F and | F 1 ∩ ⋯ ∩ F k | ( mod p ) ∈ L for any collection of k distinct sets from F . If n > ( r + 1 ) ( s − 2 r + 2 ) , then | F | ≤ ( k − 1 ) ∑ i = s − 2 r + 1 s n − 1 i . The first result improves a result of Frankl about families with given difference sizes between subsets and the second result gives an improvement to a theorem by Grolmusz–Sudakov and a theorem by W. Cao, K.W. Hwang, and D.B. West.