  In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq
\mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus
\{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding
almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of
$\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on
the colouring.
  Among other results, we characterise those $(S,s_0)$ with $S\subseteq
\mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an
infinite monochromatic arithmetic progression contains an almost monochromatic
similar copy of $(S,s_0)$. We also show that if $S\subseteq \mathbb{Z}^d$ and
$s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite
colouring of $\mathbb{R}^d$ without a similar monochromatic copy of
$\mathbb{Z}^d$ contains an almost monochromatic similar copy of $(S,s_0)$.
  Further, we propose an approach of finding almost-monochromatic sets that
might lead to a non-computer assisted proof of $\chi(\R^2)\geq 5$.
