  We study Tur\'an 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 $F\in\mathcal{F}$, and the Tur\'an 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 Tur\'an 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.
