OPEN
Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.
In other words, there is no balanced colouring. A conjecture of Erdős and Gyárfás
[ErGy99], who proved it for $r=3$ and $r=4$ (and observered it is false for $r=2$), and showed this property fails for infinitely many $r$ if we replace $r^2+1$ by $r^2$.
