  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 $|E_{RB}|$ 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 $|E_{bi}|$ 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 $|E_{RB}|=|E_{bi}| = p{n\choose 2}$, ($p\in [0,1)$), 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 is
strongly connected to Tur\'an-type extremal graph theory.
