Turán and Ramsey-type results for unavoidable subgraphs

Abstract

We study Turán and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\em $\epsilon$-balanced} if each color class contains at least an $\epsilon$-proportion of its edges. Given a family $\mathcal{F}$ of edge-colored graphs, the Ramsey function $R(\epsilon ,\mathcal{F})$ is the smallest $n$ for which any $\epsilon$-balanced $K_n$ must contain a copy of an $F\in \mathcal{F}$, and the Turán function $\mathrm{ex}(\epsilon ,n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex $\epsilon$-balanced graph which avoids all of $\mathcal{F}$. In this paper, we consider this Turán 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.