Monochromatic set
= Monochromatic set
{title2=$c\restriction A\text{ is constant}$}
= Monochromatic
{synonym}
A <subset> $A\subseteq X$ is monochromatic for a <finite coloring> $c:X\to C$ if there is a color $i\in C$ such that
$$
c(a)=i\qquad\text{for every }a\in A.
$$
Equivalently, the <restriction of a function> $c\restriction A$ is a <constant function>, or $A$ lies in one <colour class> $c^{-1}(i)$. The word describes agreement of colors; it does not require $A$ to equal an entire <colour class>.
For example, color the <integers> by <integer parity>, using red for even and blue for odd. The <set> $\{2,4,8\}$ is <monochromatic>, whereas $\{2,3\}$ is not. If the colored objects are <edges>, a <monochromatic> <triangle in a graph> means that its three <edges> form a <monochromatic set>. In a coloring of $r$-element <subsets> of a <set> $S$, a <homogeneous set for a colouring> $H\subseteq S$ means that the collection of all $r$-element <subsets> of $H$ is <monochromatic>. For pairs, this is a <complete graph> on $H$ whose <edges> all have one color, rather than a coloring of the <vertices> in $H$.