A note on unavoidable patterns in locally dense colourings

Abstract

We show that there is a constant $C$ such that for every $\epsilon >0$ any $2$-coloured $K_n$ with minimum degree at least $n/4+\epsilon n$ in both colours contains a complete subgraph on $2t$ vertices where one colour class forms a $K_{t,t}$, provided that $n\ge {\epsilon }^{-Ct}$. Also, we prove that if $K_n$ is $2$-coloured with minimum degree at least $\epsilon n$ 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 Kamčev and Müyesser.