  Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the
following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$
to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in
$K_n$, there is some $u\in V(T)$ such that every edge in $T$ has a different
colour in $f_u$.
  The study of this function was initiated by Alon and Ben-Eliezer. They
characterized the family of graphs $H$ for which $g(n,H)$ is bounded and asked
whether it is true that for every other graph $g(n,H)$ is polynomial. We show
that this is not the case and characterize the family of connected graphs $H$
for which $g(n,H)$ grows polynomially. Answering another question of theirs, we
also prove that for every $\varepsilon>0$, there is some $r=r(\varepsilon)$
such that $g(n,K_r)\geq n^{1-\varepsilon}$ for all sufficiently large $n$.
  Finally, we show that the above problem is connected to the
Erd\H{o}s-Gy\'arf\'as function in Ramsey Theory, and prove a family of special
cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each
fixed $r$ the complete $r$-uniform hypergraph $K_n^{(r)}$ can be edge-coloured
using a subpolynomial number of colours in such a way that at least $r$ colours
appear among any $r+1$ vertices.
