Toppling kings in multipartite tournaments by introducing new kings

Let and be two n-tuples of nonnegative integers. An all-4-kings n-partite tournament T(V₁,V₂,…V_n) is said to have a -property if there exists an n-partite tournament T₁(W₁,W₂,…,W_n) such that for each i∈{1,…,n}:

(1)

V_i⊆W_i;

(2)

exactly t_i 4-kings of V_i are not 4-kings in T₁;

(3)

exactly c_i 4-kings of W_i are not vertices of V_i.

We describe all pairs such that there exists an n-partite tournament having -property.