Unavoidable patterns in locally balanced colourings

Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$ , $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph. When $\varepsilon \leq 1/4$ , this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$ . Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$ , and let the blue colour class consist of the edges between $(V_1,V_2)$ , $(V_2,V_3)$ , $(V_3,V_4)$ , and the edges contained inside $V_2$ and inside $V_3$ . Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$ ) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$ .