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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 suchthat every edge is colored with at least one color and the number of bicolorededges $|E_{RB}|$ is given. The aim is to find the maximal size $f$ of amonochromatic clique which is guaranteed by such a coloring. Analogously, inthe second problem we consider semicomplete digraph on $n$ vertices such thatthe number of bi-oriented edges $|E_{bi}|$ is given. The aim is to bound thesize $F$ of the maximal transitive subtournament that is guaranteed by such adigraph.  Applying probabilistic and analytic tools and constructive methods we showthat if $|E_{RB}|=|E_{bi}| = p{n\choose 2}$, ($p\in [0,1)$), then $f, F <C_p\log(n)$ where $C_p$ only depend on $p$, while if $m={n \choose 2} -|E_{RB}| <n^{3/2}$ then $f= \Theta (\frac{n^2}{m+n})$. The latter case isstrongly connected to Tur\'an-type extremal graph theory.

Density version of the Ramsey problem and the directed Ramsey problem