Density version of the Ramsey problem and the directed Ramsey problem

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges $\mathrm{\mid }{E}_{RB}\mathrm{\mid }$ is given. The aim is to find the maximal size $f$ of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on $n$ vertices such that the number of bi-oriented edges $\mathrm{\mid }{E}_{bi}\mathrm{\mid }$ is given. The aim is to bound the size $F$ of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if $\mathrm{\mid }{E}_{RB}\mathrm{\mid }=\mathrm{\mid }{E}_{bi}\mathrm{\mid }=p\left(\genfrac{}{}{0px}{}{n}{2}\right)$, ($p\in [0,1)$), then $f, F < C_p\log(n)$where$C_p$onlydependon$p$,whileif$m={n \choose 2} - |E_{RB}| <n^{3/2}$then$f= \Theta (\frac{n^2}{m+n})$. The latter case is strongly connected to Tur'an-type extremal graph theory.