A note on unavoidable patterns in locally dense colourings

TL;DR

This paper investigates unavoidable patterns in 2-coloured complete graphs with high minimum degree, proving the existence of specific bipartite subgraphs and natural colourings, thus answering recent open questions.

Contribution

It establishes tight bounds for the presence of certain bipartite subgraphs and natural colourings in dense 2-coloured complete graphs, advancing understanding of unavoidable patterns.

Findings

Existence of a complete bipartite subgraph $K_{t,t}$ under high minimum degree conditions.

Presence of natural colourings in graphs with minimum degree proportional to $n$.

Results are tight up to a constant factor, answering recent open questions.

Abstract

We show that there is a constant $C$ such that for every $\varepsilon>0$ any $2$-coloured $K_n$ with minimum degree at least $n/4+\varepsilon n$ in both colours contains a complete subgraph on $2t$ vertices where one colour class forms a $K_{t,t}$, provided that $n\geq \varepsilon^{-Ct}$. Also, we prove that if $K_n$ is $2$-coloured with minimum degree at least $\varepsilon n$ in both colours then it must contain one of two natural colourings of a complete graph. Both results are tight up to the value of $C$ and they answer two recent questions posed by Kam\v{c}ev and M\"{u}yesser.