On a generalized anti-Ramsey problem

For positive integers p, q(1), q(2), a coloring of E(K-n) is called a (p, q(1), q(2))-coloring if the edges of every K-p receive at least q(1) and at most q(2) colors. Let R(n, p, q(1), q(2)) denote the maximum number of colors in a (p,q(1),q(2))-coloring of K-n. Given (p,q(1)) we determine the largest q(2) such that all (p,q(1),q(2))-colorings of E(K-n) have at most O(n) colors and we determine R(n,p,q(1),q(2)) asymptotically when it is of order equal to n(2). We give several bounds and constructions.