Given a graph H, let g(n,H) denote the smallest k for which the
following holds. We can assign a k-colouring fv of the edge set of Kn
to each vertex v in Kn with the property that for any copy T of H in
Kn, there is some u∈V(T) such that every edge in T has a different
colour in fu.
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 ε>0, there is some r=r(ε)
such that g(n,Kr)≥n1−ε 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 Kn(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.Comment: 12 page