Unavoidable patterns in locally balanced colourings

Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\epsilon n$ red and $\epsilon n$ blue neighbours? In this paper, we investigate this question and its multicolour variant. For instance, we show that any such graph contains a $t$-blow-up of an $\mathit{\text{alternating}}\mathit{\text{4-cycle}}$ with $t=\mathrm{\Omega }(\log n)$.