A new class of Ramsey–Turán problems

We introduce and study a new class of Ramsey–Turán problems, a typical example of which is the following one: > 0 and G be a graph of sufficiently large order n with minimum degree δ ( G ) > 3 n / 4 . If the edges of G are colored in blue or red, then for all k ∈ [ 4 , ⌊ ( 1 / 8 − ε ) n ⌋ ] , there exists a monochromatic cycle of length k .