Avoiding abelian squares with three colours in the continuous setting

Source abstract

An $n$-colouring of the real line is a Lebesgue measurable function $f \colon \mathbb{R} \to A$, where $A$ is a set of $n$ colours. An abelian square for $f$ is a finite interval $U\subseteq \mathbb{R}$ where every colour shows up in the left and in the right half of $U$ with exactly the same Lebesgue measure. It is known that every $2$-colouring contains an abelian square. On the other hand, Alon, Grytczuk, Laso{ń}, and Michałek (2009) proved that there exist $4$-colourings that avoid abelian squares, while the question for $3$-colourings remained open. We construct a $3$-colouring that avoids abelian squares, disproving a conjecture of Laso{ń} (2015). However, in contrast to the case of 4-colourings, the set of 3-colourings that avoid abelian squares is meagre. We show this using an argument based on the KKM lemma.