Liu–Morris–Prince logarithmic-factor conjecture for multicolour connectivity

Let $f(n,r,s)$ be the smallest integer such that every $r$-edge-colouring of the complete graph $K_n$ contains a set of at least $f(n,r,s)$ vertices whose induced subgraph is connected using at most $s$ colours. Here $r$ is the number of colours, $s$ is the allowed number of colours, and $n$ is sufficiently large.

Logarithmic-factor conjecture. For any

$$1\leq s\leq \frac{\sqrt r}{\log r}$$

and $n$ large enough,

$$f(n,r,s)=\frac{s^2(\log r)^{1-o_r(1)}}{r}\cdot n.$$

The conjecture asks whether the logarithmic factor in the known lower bound is necessary throughout this range of $s$. It is motivated by the case $s\approx \log\log r$, where the authors show that a $(\log r)^{1-o_r(1)}$ factor is necessary; the general range stated above remains open.