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Fix positive integers $r,k,m$ with $m\ge r$. There is an integer $n\ge m$, depending only on $r,k,m$, such that every finite coloringhas a subset $H\subseteq [n]$ of size $m$ for which the restriction of a function $c$ to $\left(\genfrac{}{}{0ex}{}{H}{r}\right)$ is a constant function. Here $[n]=\{1,\dots ,n\}$, and $\left(\genfrac{}{}{0ex}{}{S}{r}\right)$ denotes the collection of all $r$-element subsets of $S$, rather than its binomial coefficient cardinality. The notation $[n]⟶(m{)}_k^r$ abbreviates this property. Thus $H$ is a homogeneous set for a colouring; when $r>1$, the monochromatic set consists of its $r$-element subsets, not its individual elements. This is the finite assertion in Ramsey's original Theorem B, p. 267.

For $r=1$, this is the pigeonhole principle, with sharp threshold $n=k(m-1)+1$. For instance, every red-blue coloring of five objects has three of one color; four objects can split two and two. For $r=2$, $k=2$ and $m=3$, the sharp threshold is six, as shown by the Ramsey number for triangles. Coloring pairs makes this a problem about edges of a complete graph. For larger $r$, the analogous viewpoint uses a hypergraph.

The finite Ramsey theorem lets an existence argument discard most of a large finite set while retaining a prescribed amount of uniform structure. Its threshold applies to every coloring; one does not need an assumption that colors were assigned randomly. A diagonal compactness argument deduces finite existence from Ramsey's theorem, but does not by itself identify the smallest threshold. For pairs, a Ramsey number makes that optimization explicit.

Let $X=\{v_1,\dots ,v_m\}$ be the vertices of a regular polygon, indexed cyclically so that $v_i\mapsto v_{i+1}$ is an isometry. We first prove the product lemma needed for the cyclic-symmetry argument.

Suppose $A\subseteq X$ has the property that, for every number of colors, a finite witness forces a copy of $X$ on which the vertices corresponding to $A$ are monochromatic. We claim that for every $N,k$ there is a finite witness forcing an $A$-invariant copy of $X^N$. Proceed by mathematical induction on $N$. The case $N=1$ is the assumption. For the step, choose a finite set $S$ that forces an $A$-invariant $X^{N-1}$ for a coloring with $k^{|X|-|A|+1}$ colors, and choose $T$ that forces an $A$-monochromatic $X$ for a coloring with $k^{|S|}$ colors.

Given a $k$-coloring $c$ of $S\times T$, color $t\in T$ by its complete vector $(c(s,t){)}_{s\in S}$. In the resulting copy of $X$ in $T$, that vector is the same for every member of $A$. Each $s\in S$ can therefore be colored by the vector consisting of $c(s,a)$ for one $a\in A$ and the colors $c(s,x)$ for $x\in X\setminus A$. Applying the choice of $S$ gives an $A$-invariant copy of $X^{N-1}$, and its product with the chosen copy of $X$ in $T$ is the desired $A$-invariant copy of $X^N$. This proves the lemma rather than assuming it.

We now prove by induction on $r$ that a finite witness forces a copy of $X$ whose first $r$ vertices have one color. The case $r=1$ is trivial. For $r=2$, use that a line segment is a Euclidean Ramsey set; each occurrence of a segment congruent to $v_1{v}_2$ can be extended to a copy of the regular polygon, and only finitely many extensions are needed for a finite witness.

Assume the assertion for $r$, and put $A=\{v_1,\dots ,v_r\}$. The product lemma lets us work inside an $A$-invariantly colored copy of $X^N$, with $N$ as large as required. For an increasing $(m-1)$-element subset $H=\{a_1<\cdots <a_{m-1}\}$ of $[N]$ and $1\le i\le r$, define a word $w_i(H)$ by putting $v_{i+j}$ in coordinate $a_j$, with subscripts read modulo $m$, and putting $v_1$ in every other coordinate. Color $H$ by the $r$-tupleBy the Finite Ramsey theorem, for sufficiently large $N$ there are coordinates $b_1<\cdots <b_m$ on which all $(m-1)$-subsets have the same tuple color.

For $1\le i\le r+1$, let $z_i$ be the word whose entries in coordinates $b_1,\dots ,b_m$ arecyclically, and whose other entries are $v_1$. For $i\le r$, the word $z_i$ differs from $w_i(\{b_2,\dots ,b_m\})$ only in coordinate $b_1$, where the two entries $v_i,v_1$ lie in $A$. Similarly $z_{i+1}$ differs from $w_i(\{b_1,\dots ,b_{m-1}\})$ only in coordinate $b_m$, again by two members of $A$. The $A$-invariance and the homogeneous choice of the $b_j$ therefore giveHence $z_1,\dots ,z_{r+1}$ have one color.

The cyclic words $z_1,\dots ,z_m$ form an isometric copy of $\sqrt{m}X$: in each of the $m$ varying coordinates the cyclic shift is an isometry, so every squared distance is multiplied by $m$. Rescaling the finite witness by $1/\sqrt{m}$ gives a copy of $X$. The induction reaches $r=m$, proving that every regular polygon is a Euclidean Ramsey set.

For three consecutive vertices $p_0,p_1,p_2$ of a regular $M$-gon scaled to have side length one,Since the polygon is Euclidean Ramsey, choosing $M$ large enough proves that $\{0,1,2\}$ is an approximately Euclidean Ramsey set.

In fact every finite $X\subseteq {\mathbb{R}}^d$ is approximately Ramsey. First perturb each coordinate of each point onto a sufficiently fine lattice $\delta \mathbb{Z}$, changing all pairwise distances by less than $\epsilon /2$. For each coordinate, map the finitely many required integers to consecutive vertices of a very large regular polygon, scaled so that one angular step has arc length $\delta$. If the step angle is $\theta$, the chord replacing a difference of $q$ lattice steps has lengthas $\theta \to 0$, uniformly over the finitely many differences involved. Taking the orthogonal Cartesian product of $d$ such polygons therefore embeds the perturbed points with a further distance error below $\epsilon /2$. Each polygon is Euclidean Ramsey, and their product is Euclidean Ramsey by the product theorem for Euclidean Ramsey sets. A monochromatic copy of that product contains the corresponding approximate copy of $X$, completing the proof.

Suppose the Finite Ramsey theorem failed for fixed positive integers $r,k,m$. For every $n$ choose a $k$-coloring $c_n:[n{]}^{(r)}\to [k]$ with no monochromatic $m$-set. There are only finitely many colorings of $[r{]}^{(r)}$, so an infinite subsequence of the $c_n$ agrees there. Pass to a further infinite subsequence agreeing on $[r+1{]}^{(r)}$, and continue. The diagonal argument produces compatible colorings $d_N:[N{]}^{(r)}\to [k]$ such that $d_{N+1}{|}_{[N{]}^{(r)}}=d_N$ and no $d_N$ has a monochromatic $m$-set.

Define $d(F)=d_N(F)$ whenever $N\ge \max F$. Compatibility makes this a well-defined finite coloring of $[\mathbb{N}{]}^{(r)}$. By Ramsey's theorem it has an infinite monochromatic set, whose first $m$ elements contradict the defining property of a sufficiently large $d_N$. This compactness argument proves the finite statement.

Ramsey theory studies configurations that every finite coloring of a sufficiently large structure must contain. Its striking message is that some order is unavoidable even when no organized construction was used. A basic example is the Ramsey number for triangles: color every edge of a complete graph on six vertices red or blue, in any way, and some triangle in a graph has all three edges the same color. Five vertices do not suffice.

The objects receiving colors matter. In the Ramsey number for triangles, colors belong to edges, which represent all unordered pairs of distinct vertices; the vertices themselves have no colors. In the finite Ramsey theorem, colors can instead belong to all $r$-element subsets of a finite set. A monochromatic set or homogeneous set for a colouring expresses agreement among the appropriate colored objects.

Why is this useful? Ramsey theory provides existence guarantees in combinatorics that survive every possible assignment of a fixed finite number of colors. The Ramsey theorem turns this into a structural dichotomy: every sufficiently large graph contains a large clique or a large independent set. In arithmetic, the Van der Waerden theorem guarantees monochromatic arithmetic progressions. The beauty is the universal quantifier: an adversary can choose every color and still cannot prevent the promised configuration.

These guarantees concern precisely specified configurations under stated hypotheses. They do not justify selecting arbitrary patterns in empirical data after seeing the data, or attributing statistical significance to a selected pattern. A mathematical guarantee of some monochromatic configuration alone says nothing about the empirical meaning of that configuration.

The subject's name comes from F. P. Ramsey's ^[ref]. The ^[ref] records receipt on 28 November 1928 and presentation on 13 December 1928. Ramsey introduced the Ramsey theorem for r-sets and finite Ramsey theorem as tools for a problem in mathematical logic, while recognizing their independent interest in combinatorics. He also distinguished proving that a finite threshold exists from finding its smallest value, the quantitative question captured by a Ramsey number.