There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a finite coloring of $\mathbb Z^d$ or $\mathbb R^d$, there is a homothetic (i.e., scaled and translated) copy of any finite configuration $S$. Motivated by this problem, I wonder if similar statements hold if instead we require that in a configuration $(S,s_0)$ all points in $S$ are monochromatic, while $s_0$ has a different color from the rest.

Obviously, we need to impose some conditions on the coloring and the configuration. About the coloring, I only want to demand that it is non-monochromatic, i.e., not all points of the space are colored with the same color. About $(S,s_0)$, I want to require that $s_0\notin conv(S)$, i.e., $s$ is not in the convex hull of some points from $S$, as then we might not have a solution if the "first" half of the space is red, while the "second" half is blue.

Is there always an almost monochromatic copy (homothetic or isometric) of any finite $(S,s_0)$ with $s_0\notin conv(S)$ in a non-monochromatic finite coloring of $\mathbb R^d$?

Note that the answer is no for $\mathbb Z$ if $s_0=0$ and $S=\{1,2\}$ as shown by coloring odd numbers red and even numbers blue. This particular configuration, however, is easy to find in $\mathbb Q$.

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