Assume on the contrary that there is a coloring for which any triangle with side lengths $1$, $2$, $\sqrt{3}$ has at least one red vertex and one blue vertex. Consider an equilateral triangle $ABC$ with side length $2$.

There are at least two vertices among $A$, $B$, $C$ with the same color, let them be $B$, $C$ with red color.
Let $D$, $E$ be the midpoints of $AB$, $AC$, respectively, and let $D'$, $E'$ be their reflections with respect to the line $BC$. The triangles $BDC$, $BEC$, $BD'C$, $BE'C$ all have side lengths $1$, $2$, $\sqrt{3}$, and so the vertices $D$, $E$, $D'$, $E'$ are of blue color. However, the triangle $DD'E$ has side lengths $1$, $2$, $\sqrt{3}$ but all of its vertices are of blue color, which is a contradiction.