Weighted Turan Problems with Applications

Suppose the edges of $K_n$ are assigned weights by a weight function $w$. We define the {\em weighted extremal number} $$\mathrm{e}\mathrm{x}(n,w,F):=\max \{w(G)\mid G\subseteq K_n,\text{ and }G\text{ is }F\text{-free}\}$$ where $w(G):={\sum }_{e\in E(G)}w(e)$. In this paper we study this problem for two types of weights $w$, each of which has an application. The first application is to an extremal problem in a complete multipartite host graph. The second application is to the maximum rectilinear crossing number of trees of diameter 4.