  In this paper, we address problems related to parameters concerning edge
mappings of graphs. Inspired by Ramsey's Theorem, the quantity $m(G, H)$ is
defined to be the minimum number $n$ such that for every $f: E(K_n) \rightarrow
E(K_n)$ either there is a fixed copy of $G$ with $f ( e) = e$ for all $e\in
E(G)$, or a free copy of $H$ with $f( e) \notin E(H)$ for all $e\in E(H)$. We
extend many old results from the 80's as well as proving many new results. We
also consider several new interesting parameters with the same spirit.
