I recently came across this paper: http://borisalexeev.com/pdf/foxgraham.pdf
"On Minimal Colorings Without Monochromatic Solutions To a Linear Equation"
Can someone explain in clearer terms what it means for a linear equation to not have a monochromatic solution with minimal colorings?
Let $C$ be a set of colors. Suppose every integer is colored with one of the colors in $C$. Formally, this is just a mapping $\chi: \Bbb Z\to C$ that says, for each integer $i$, the color $\chi(i)$ of the integer $i$. This mapping is called a coloring of the integers. One could similarly consider colorings of $\Bbb R$ or whatever.
Now consider a linear equation, say $$x+y=z.$$ A solution to the equation is a triple $\langle x, y, z\rangle$ of three numbers for which $x+y=z$. If we have a coloring $\chi$ in mind, a monochromatic solution is a solution that has all three of $x,y,z$ the same color; that is $\chi(x) = \chi(y) = \chi(z)$.
Given a coloring, we can ask if the equation has a monochromatic solution; or given some set of colors we can ask if the equation has a monochromatic solution for every coloring with those colors.
A theorem of Schur solves the problem completely for the equation $x+y=z$, which has a monochromatic solution in integers for any coloring with any finite number of colors. (Clearly, there may not be a monochromatic solution if the number of colors is infinite.) For different equations, the answer is different.
Suppose some equation $E$ has a monochromatic solution for every possible coloring of the integers with $p$ different colors, but there is some coloring $\chi$ with $p+1$ colors for which there is no monochromatic solution to $E$.
which is your question. (The definition in the paper is at the top of page 2, labeled Definition.)