Unavoidable patterns in locally balanced colourings

TL;DR

This paper investigates unavoidable patterns in two-coloured complete graphs where each vertex has a balanced number of red and blue neighbours, demonstrating the existence of large blow-ups of alternating 4-cycles.

Contribution

It establishes that such balanced colourings necessarily contain large blow-ups of specific patterns, extending understanding of structural properties in coloured graphs.

Findings

Existence of a t-blow-up of an alternating 4-cycle with t = Omega(log n)

Balanced colourings contain large structured patterns

Results extend to multicolour variants

Abstract

Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ 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 \textit{alternating 4-cycle} with $t = \Omega(\log n)$.