Finite coloring

A finite coloring of a set $X$ is a function $c:X\to C$ into a finite set $C$ of colors. A subset is monochromatic when the restriction of $c$ to it is constant.

The colour profile of a length-$N$ block $b_1,\dots ,b_N$ under a finite colouring $c$ is the ordered tuple $(c(b_1),\dots ,c(b_N))$. There are at most $r^N$ profiles when $c$ uses at most $r$ colours. Equally profiled blocks reproduce every colour configuration at the same relative positions; this converts a colouring of points into a finite colouring of blocks in the colour-focusing proof of Van der Waerden theorem.

A colour class of a finite colouring $c:X\to \{1,\dots ,r\}$ is a fibre $c^{-1}(i)$. Each colour class is a monochromatic set, and these fibres give a finite set partition of $X$ after empty fibres are omitted.

For $b>1$ and an integer $q\ge 2$, assign $x\ge 1$ the color $\lfloor {\log }_{b}x\rfloor \mathrm{mod}q$. The floor function partitions the domain into half-open geometric bins. If $1<{\log }_b(y/x)<q-1$, their bin indices differ by an integer in $\{1,\dots ,q-1\}$, so their colors differ. For example, $b=3/2$ and $q=3$ separates every ratio in $[1.9,2]$. Composing this finite coloring with another logarithm can obstruct monochromatic multiplicative configurations whose logarithms have such ratios.

A refinement of a finite coloring assigns different new colors whenever the original colors differ. A set monochromatic for the refinement is therefore monochromatic for the original coloring. Given colorings $\chi ,\psi$, the product coloring $n\mapsto (\chi (n),\psi (n))$ refines both. This makes it possible to impose finitely many coloring obstructions simultaneously.

A subset $A\subseteq X$ is monochromatic for a finite coloring $c:X\to C$ if there is a color $i\in C$ such thatEquivalently, the restriction of a function $c\upharpoonright A$ 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 $H$.