  In this note we study graphs $G_r$ with the property that every colouring of
$E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$
colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps
in the synthesis of a $2$-colour Ramsey graph $G_1\longrightarrow H$. (The
corresponding notion of Ramsey-type numbers was introduced by Erd\"os, Hajnal
and Rado in 1965 and subsequently studied by Erd\"os and Szemer\'edi in 1972).
For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and
establish Ramsey-infiniteness. From the structural point of view, we
characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to
be the graph property of containing a cycle; we then use it to progress towards
a constructive description of that class by proving both a reduction and an
extension theorem.
