Edge mappings of graphs: Ramsey type parameters

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 $f:E(K_n)\to E(K_n)$ 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 $f(e)\mathrm{\notin }E(H)$ 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.