Extremal Problems Concerning Transformations of the Set of Edges of the Complete Graph

Let E_m denote the set of edges of the complete graph on m vertices K_m, and let f : E_m → E_m be a function. A subgraph G = (V (G), E (G)) of K_m is called f-fixed if f(e) = e for all e ∈ E (G) and f-free if f(e) ∉ E(G) for all e ∈ E(G). For two finite graphs G, H we define m(G,H)=max{m:∃f:Em→EmsuchthatnocopyofGinKmisf-fixedandnocopyofHinKmisf-free} If m > 2 we define l(m,H)=max{l:∃f:Em→Em,f(e)≠eforledgese∈EmandnocopyofHinKmisf-free} In this paper we investigate the functions m(G, H) and l(m, H). We determine m(G, H) precisely for some families of graphs and estimate the asymptotic behaviour of l(m, H) for fixed H as m tends to infinity. Some of the results are generalised to functions defined on the set of edges of a hypergraph.