A canonical Ramsey theorem with list constraints in random (hyper-)graphs

Abstract

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K_n^{(k)}$ gives rise to copies of $H$ that exhibit certain colour patterns. We are interested in sparse random versions of this result and the threshold at which the random $k$-graph ${\mathbf{G}}^{(k)}(n,p)$ inherits the canonical Ramsey properties of $K_n^{(k)}$. Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that $H$ is a (2-uniform) even cycle.

18 pages, final version to appear in The Electronic Journal of Combinatorics (E-JC). An extended abstract introducing our result here (focusing just on the case of graphs) appeared in the proceedings of the XII Latin-American Algorithms, Graphs and Optimization Symposium (LAGOS 2023)