A new type of Ramsey-Turan problems

TL;DR

This paper introduces a novel class of Ramsey-Turan problems, demonstrating that large graphs with high minimum degree and two-color edgeings contain monochromatic cycles of various lengths.

Contribution

It defines a new problem type and establishes conditions under which monochromatic cycles of certain lengths must exist in two-colored large graphs.

Findings

Monochromatic cycles of length k exist for 3<k<(1/8-c)n in large graphs with high minimum degree.

The study extends Ramsey-Turan theory to new cycle length ranges.

Provides bounds and conditions for monochromatic cycle existence.

Abstract

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.