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 $E$.

Then this coloring $\chi$ is a minimal coloring without a monochromatic solution to the equation

which is your question. (The definition in the paper is at the top of page 2, labeled **Definition**.)