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 $u\in V(T)$ 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 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 $K_n^{(r)}$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2023
- DOI:
- 10.48550/arXiv.2304.12260
- arXiv:
- arXiv:2304.12260
- Bibcode:
- 2023arXiv230412260J
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 12 pages