  Motivated by the investigation of sharpness of thresholds for Ramsey
properties in random graphs, Friedgut, Kohayakawa, R\"odl, Ruci\'nski and
Tetali introduced two variants of a single-player game whose goal is to colour
the edges of a~random graph, in an online fashion, so as not to create a
monochromatic triangle. In the two-round variant of the game, the player is
first asked to find a triangle-free colouring of the edges of a random graph
$G_1$ and then extend this colouring to a triangle-free colouring of the union
of $G_1$ and another (independent) random graph $G_2$, which is disclosed to
the player only after they have coloured $G_1$. Friedgut et al.\ analysed this
variant of the online Ramsey game in two instances: when $G_1$ has
$\Theta(n^{4/3})$ edges and when the number of edges of $G_1$ is just below the
threshold above which a random graph typically no longer admits a triangle-free
colouring, which is located at $\Theta(n^{3/2})$.
  The two-round Ramsey game has been recently revisited by Conlon, Das, Lee and
M\'esz\'aros, who generalised the result of Friedgut at al.\ from triangles to
all strictly $2$-balanced graphs. We extend the work of Friedgut et al.\ in an
orthogonal direction and analyse the triangle case of the two-round Ramsey game
at all intermediate densities. More precisely, for every $n^{-4/3} \ll p \ll
n^{-1/2}$, with the exception of $p = \Theta(n^{-3/5})$, we determine the
threshold density $q$ at which it becomes impossible to extend any
triangle-free colouring of a typical $G_1 \sim G_{n,p}$ to a triangle-free
colouring of the union of $G_1$ and $G_2 \sim G_{n,q}$. An interesting aspect
of our result is that this threshold density $q$ `jumps' by a polynomial
quantity as $p$ crosses a `critical' window around $n^{-3/5}$.
