In a colouring of R da pair (S, s 0) with S⊆ R d and with s 0∈ S is almost-monochromatic if
S {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 R d, Z d, and of Q under some
restrictions on the colouring. Among other results, we characterise those (S, s 0) with S⊆ Z
for which every finite colouring of R without an infinite monochromatic arithmetic progression
contains an almost-monochromatic similar copy of (S, s 0). We also show that if S⊆ Z d and …

In a colouring of Rd a pair (S, s0) with S⊆ Rd and with s0∈ S is almostmonochromatic if
S\{s0} is monochromatic but S is not. We consider questions about finding almost-
monochromatic similar copies of pairs (S, s0) in colourings of Rd, Z d, and of Q under some
restrictions on the colouring. Among other results, we characterise those (S, s0) with S⊆ Z
for which every finite colouring of R without an infinite monochromatic arithmetic progression
contains an almost-monochromatic similar copy of (S, s0). We also show that if S⊆ Zd and …