We study Turan and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {em $varepsilon$-balanced} if each color class contains at least an $varepsilon$-proportion of its edges. Given a family $mathcal{F}$ of edge-colored graphs, the Ramsey function $R(varepsilon, mathcal{F})$ is the smallest $n$ for which any $varepsilon$-balanced $K_n$ must contain a copy of an $Finmathcal{F}$, and the Turan function $mathrm{ex}(varepsilon, n, mathcal{F})$ is the maximum number of edges in an $n$-vertex $varepsilon$-balanced graph which avoids all of $mathcal{F}$. In this paper, we consider this Turan function for several classes of edge-colored graphs, we show that the Ramsey function is linear for bounded degree graphs, and we prove a theorem that gives a relationship between the two parameters.
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