The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given graph H, if n is sufficiently large then any colouring of the edges of Kn gives rise to copies of H that exhibit certain colour patterns, namely monochromatic, rainbow or lexicographic. We are interested in sparse random versions of this result and the threshold at which the random graph G(n,p) inherits the canonical Ramsey properties of Kn. 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 an even cycle.

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum d-degree threshold for loose Hamiltonicity relative to the random k-uniform hypergraph Hk(n, p) coincides with its dense analogue whenever p ≥ n−(k-1)/2+o(1). The value of p is approximately tight for d > (k + 1)/2. This is particularly interesting because the dense threshold itself is not known beyond the cases when d ≥ k - 2.