{"schema":"vela.problem-packet.v0.1","problem":1105,"statement":"The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours).Let $C_k$ be the cycle on $k$ vertices. Is it true that\\[\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)?\\]Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to\\[\\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right)\\]where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}