An analogue of the Erdős–Ko–Rado theorem for weak compositions

Let N 0 be the set of non-negative integers, and let P ( n , l ) denote the set of all weak compositions of n with l parts, i.e., P ( n , l ) = { ( x 1 , x 2 , … , x l ) ∈ N 0 l : x 1 + x 2 + ⋯ + x l = n } . For any element u = ( u 1 , u 2 , … , u l ) ∈ P ( n , l ) , denote its i th-coordinate by u ( i ) , i.e., u ( i ) = u i . A family A ⊆ P ( n , l ) is said to be t -intersecting if | { i : u ( i ) = v ( i ) } | ≥ t for all u , v ∈ A . We prove that given any positive integers l , t with l ≥ t + 2 , there exists a constant n 0 ( l , t ) depending only on l and t , such that for all n ≥ n 0 ( l , t ) , if A ⊆ P ( n , l ) is t -intersecting then | A | ≤ n + l − t − 1 l − t − 1 . Moreover, the equality holds if and only if A = { u ∈ P ( n , l ) : u ( j ) = 0 for all j ∈ T } for some t -set T of { 1 , 2 , … , l } .