  We introduce and study a new type of Ramsey-Turan problems, a typical example
of which is the following one: let c>0 and G be a graph of sufficiently large
order n with minimum degree >3n/4. If the edges of G are colored in blue or
red, then for every 3<k<(1/8-c)n, there exists a monochromatic cycle of length
k.
