  We prove Tur\'an-type theorems for two related Ramsey problems raised by
Bollob\'as and by Fox and Sudakov. First, for $t \ge 3$, we show that any
two-colouring of the complete graph on $n$ vertices that is $\delta$-far from
being monochromatic contains an \emph{unavoidable $t$-colouring} when $\delta
\gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a
clique of order $2t$ in which one colour forms either a clique of order $t$ or
two disjoint cliques of order $t$. Next, for $ t\ge 3$, we show that any
tournament on $n$ vertices that is $\delta$-far from being transitive contains
an \emph{unavoidable $t$-tournament} when $\delta \gg n^{-1/\lceil t/2
\rceil}$, where an unavoidable $t$-tournament is the blow-up of a cyclic
triangle obtained by replacing each vertex of the triangle by a transitive
tournament of order $t$. Conditional on a well-known conjecture about bipartite
Tur\'an numbers, both results are sharp up to implied constants and hence
determine the order of magnitude of the corresponding off-diagonal Ramsey
numbers.
