Entropy Transference for Rainbow-$H$-Free Colourings of Random Graphs

Abstract

Let $H$ be a fixed graph with $q=e(H)\ge 3$ and containing two adjacent edges, and let $\ell \ge q$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Below a sufficiently small constant multiple of this scale, almost all host edges may be coloured freely; above a sufficiently large constant multiple, the exponential counting rate is governed exactly by a deterministic template-entropy optimisation on complete graphs. An exact Hall-palette inequality evaluates this rate throughout the universal range $q\le \ell \le (q-1{)}^{q/(q-2)}$, where the dense-side base is $q-1$, and a robust form yields counting stability below the endpoint. For arbitrary fixed $\ell$, deletion-profile bounds determine the first-order many-colour behaviour and characterise when the $(q-1)$-colour rate persists for every fixed number of colours. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph and provides a general mechanism for transferring dense template entropy to sparse random hosts.