On locally rainbow colourings

Abstract

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 $u\in V\left(T\right)$ 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 $\epsilon >0$, there is some $r=r\left(\epsilon \right)$ such that $g(n,K_r)\ge n^{1-\epsilon }$ for all sufficiently large n.

Finally, we show that the above problem is connected to the Erdős–Gyárfás 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 $K_n^{\left(r\right)}$ 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.