Ramsey-type problem for an almost monochromatic K_4

TL;DR

This paper proves that for large enough complete graphs with edges colored using k colors, there always exists a K_4 subgraph with edges in at most two colors, establishing an exponential bound.

Contribution

It introduces the first exponential bound for the Ramsey-type problem concerning almost monochromatic K_4 subgraphs in k-edge-colored complete graphs.

Findings

Existence of a constant c such that large enough k-edge-colored K_n contains a K_4 with at most two colors.

Improvement over previous polynomial bounds by Kostochka and Mubayi.

First exponential bound established for this problem.

Abstract

In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.