On rr-cross tt-intersecting families for weak compositions

Let N0N0 be the set of non-negative integers, and let P(n,l)P(n,l) denote the set of all weak compositions of nn with ll parts, i.e., P(n,l)={(x1,x2,…,xl)∈N0l:x1+x2+⋯+xl=n}. For any element u=(u1,u2,…,ul)∈P(n,l), denote its iith-coordinate by u(i), i.e., u(i)=ui. Let l=min(l1,l2,…,lr)l=min(l1,l2,…,lr). Families Aj⊆P(nj,lj)Aj⊆P(nj,lj) (j=1,2,…,rj=1,2,…,r) are said to be rr-cross tt-intersecting if |{i∈[l]:u1(i)=u2(i)=⋯=ur(i)}|≥t for all uj∈Aj. Suppose that l≥t+2l≥t+2. We prove that there exists a constant n0=n0(l1,l2,…,lr,t)n0=n0(l1,l2,…,lr,t) depending only on ljlj’s and tt, such that for all nj≥n0nj≥n0, if the families Aj⊆P(nj,lj)Aj⊆P(nj,lj) (j=1,2,…,rj=1,2,…,r) are rr-cross tt-intersecting, then ∏j=1r|Aj|≤∏j=1rnj+lj−t−1lj−t−1. Moreover, equality holds if and only if there is a tt-set TT of {1,2,…,l}{1,2,…,l} such that Aj={u∈P(nj,lj):u(i)=0for alli∈T} for j=1,2,…,rj=1,2,…,r.