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} \[ \mathrm{ex}(n,w,F):=\max\{w(G)\mid G\subseteq K_n,\text{ and }G\text{ is }F\text{-free}\} \] 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.