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- W4367000482 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 $uin V(T)$ 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 ErdH{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." @default.
- W4367000482 created "2023-04-27" @default.
- W4367000482 creator A5011824933 @default.
- W4367000482 creator A5039637925 @default.
- W4367000482 date "2023-04-24" @default.
- W4367000482 modified "2023-09-30" @default.
- W4367000482 title "On locally rainbow colourings" @default.
- W4367000482 doi "https://doi.org/10.48550/arxiv.2304.12260" @default.
- W4367000482 hasPublicationYear "2023" @default.
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