Unavoidable patterns in locally balanced colourings

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