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:Let ε>0ε>0 and GG be a graph of sufficiently large order nn with minimum degree δ(G)>3n/4δ(G)>3n/4. If the edges of GG are colored in blue or red, then for all k∈[4,⌊(1/8−ε)n⌋]k∈[4,⌊(1/8−ε)n⌋], there exists a monochromatic cycle of length kk.