A note on edge-colourings avoiding rainbow K_4 and monochromatic K_m

TL;DR

This paper investigates the maximum number of colours in edge-colourings of complete graphs that avoid both monochromatic and rainbow complete subgraphs, providing improved bounds and discussing potential enhancements based on finite projective planes.

Contribution

It improves the upper bound on maxR(n,K_m,K_4) and explores a new approach to enhance the lower bound on maxR(n,K_4,K_4) using incidence graphs of finite projective planes.

Findings

Upper bound on maxR(n,K_m,K_4) is n^{3/2} * sqrt{2m} for m >= 3

Discussion of a method to improve the lower bound on maxR(n,K_4,K_4)

Potential connection to incidence graphs of finite projective planes

Abstract

We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,K_m,K_4) <= n^{3/2}\sqrt{2m} for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on incidence graphs of finite projective planes.