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We study Tur\'an and Ramsey-type problems on edge-colored graphs. Anedge-colored graph is called {\em $\varepsilon$-balanced} if each color classcontains 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 $\mathcal{F}$. Inthis paper, we consider this Tur\'an function for several classes ofedge-colored graphs, we show that the Ramsey function is linear for boundeddegree graphs, and we prove a theorem that gives a relationship between the twoparameters.

Tur\'an and Ramsey-type results for unavoidable subgraphs