On a generalized Anti-Ramsey problem

For positive integers p,q₁, q₂, a coloring of E(K_n) is called a (p, q₁,q₂)- coloring if the edges of every K_p receive at least q₁ and at most q₂ colors. Let R(n,p,q₁, q₂) denote the maximum number of colors in a (p,q₁,q₂)-coloring of K_n. Given (p,q₁) we determine the largest q₂ such that all (p, q₁,q₂)-colorings of E(K_n) have at most O(n) edges and we determine R(n,p,q₁, q₂) asymptotically when it is of order n². We give several additional constructions and bounds, but many questions remain open.