Gallai’s Theorem: A Detailed Exposition of Witt’s Proof

1. Introduction

A theorem of Tibor Gallai (formerly Grünwald) states that if the points in the Euclidean plane are colored with finitely many colors, then for every finite subset of the plane there is a monochromatic homothetic copy of that set. A homothetic copy of a set is its image under first a dilation and then a translation. A monochromatic copy is one in which that every point receives the same color. Another way to state Gallai’s theorem is that for every finite subset X of the plane and every number k of colors, there is another finite subset Y of the plane, such that if Y is colored by k colors then Y contains a monochromatic homothetic image of X. (The two versions are equivalent by the compactness theorem for first-order logic.)

Gallai did not publish his result. (For an explanation and additional history, one should read both [1, Ch. 42] and [2, Ch. 46].) Gallai’s theorem was first mentioned in print by Richard Rado [3]. Reviewing Rado’s paper [3] for Mathematical Reviews, Erdos wrote, “the following result of T. Grünwald is used: given any configuration S consisting of a finite number of lattice points of Euclidean space, and given a distribution of all lattice points of this space into a finite number of classes, there is at least one class which contains a configuration $S^{\prime }$ of lattice points which is similar and parallel to S.”

I first heard about Gallai’s theorem in late 2012 from Jeremy Alm, who wrote and sent me a paper about extending Gallai’s theorem [4]. I became intrigued by the theorem, tried to prove it, and eventually consulted the reference in Jeremy’s paper, Alexander Soifer’s The Mathematical Coloring Book [1, §42.3]. Soifer presents an expanded account of the proof of Gallai’s theorem published by Ernst Witt in 1952 [5]. Reviewing Witt’s paper [5] for Mathematical Reviews, Erdos wrote, “The author was unaware of a paper by R. Rado in which it is stated that the result is due to T. Grünwald (Gallai).” Soifer thought Witt’s proof was incomprehensibly brief, so he added details but did not include the proof in the 2024 edition [2]. Soifer’s proof seemed incomprehensibly brief to me, so I added even more details, enough for a purely formal and mechanical confirmation of the proof.

The theorem and its proof are written for ${\mathbb{R}}^2$, but any power of $\mathbb{R}$ can be substituted for ${\mathbb{R}}^2$. In fact, the proof applies with no changes to any ordered ring. The elements are called either points or vectors, depending on their use as either geometric or computational objects.

The theorem is proved in a slightly more precise formulation than the usual statement. From points $e_0,e_1,e_2,\dots$, a number $n\ge 3$, and a number of colors $k\ge 1$ a set $\Phi (n,k)$ is defined recursively in advance with the property that for any coloring of $\Phi (n,k)$ with k colors, $\Phi (n,k)$ contains a monochromatic homothetic copy of $\{e_0,\dots ,e_n\}$. The key idea, as in many combinatorial arguments, is to treat entire colorings as colors and to apply recursive coloring constructions within the inductive argument.

2. Definitions

Consider an arbitrary but fixed infinite sequence $S=〈e_0,e_1,e_2,\dots ,e_i,\dots 〉$ of distinct points in ${\mathbb{R}}^2$, where $e_0=(0,0)$. For every integer $n\ge 1$ let

$$S_n=\{e_0,\dots ,e_{n-1}\}.$$

(1)

For example, $S_1=\left\{e_0\right\}$, $S_2=\{e_0,e_1\}$, and $S_3=\{e_0,e_1,e_2\}$. For every set X, $\left|X\right|$ is the number of elements of X, called the cardinality of X. The cardinality of $S_n$ is n, i.e., $|S_n|=n$. For every positive integer k, a k-element set is a set X such that $\left|X\right|=k$. A function from a set Y to a k-element set X is a k-coloring of Y. For any vector $v\in {\mathbb{R}}^2$ and any two sets $V,W\subseteq {\mathbb{R}}^2$, the sumset of v (or V) and W is the set $v+W$ (or $V+W$) of vectors obtained by adding v (or any vector from V) to any vector from W, that is,

$$\begin{array}{cc}\hfill v+W& =\{v+w:w\in W\},\hfill \\ \hfill V+W& =\{v+w:v\in V,w\in W\}.\hfill \end{array}$$

Let H be the set of all functions $h:{\mathbb{R}}^2\to {\mathbb{R}}^2$ such that, for some displacement vector $a\in {\mathbb{R}}^2$ and some positive dilation scalar $\lambda \in \mathbb{R}$, we have $h\left(v\right)=a+\lambda v$ for every $v\in {\mathbb{R}}^2$, where addition and scalar multiplication are performed componentwise. H is the set of homotheties from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$, consisting of all compositions of translations and dilations, i.e.,

$$H=\{h:(\exists a\in {\mathbb{R}}^2)(\exists \lambda \in \mathbb{R})(0<\lambda ,(\forall v\in {\mathbb{R}}^2)(h\left(v\right)=a+\lambda v))\}.$$

If $V\subseteq {\mathbb{R}}^2$ and $h\in H$, then $h\left(V\right)$ is the image of V under h, i.e.,

$$h\left(V\right)=\{h\left(v\right):v\in {\mathbb{R}}^2\}.$$

For any set $V\subseteq {\mathbb{R}}^2$ and any $n\ge 2$, let $E_n\left(V\right)$ be the union of the images of $S_n$ under those homotheties that map $S_{n-1}$ into V, i.e.,

$$E_n\left(V\right)=\bigcup \{h\left(S_n\right):h\in H,h\left(S_{n-1}\right)\subseteq V\}.$$

(2)

For integers $n\ge 3$, $k\ge 1$, and $m\ge 1$, define sets $\Phi (n,k),\Delta (n,k,m)\subseteq {\mathbb{R}}^2$ by

$$\begin{array}{cc}\hfill \Phi (2,k)& =\{i{e}_1:0\le i\le k\},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \Delta (n,k,1)& =E_n(\Phi (n-1,k)),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \Delta (n,k,m+1)& =\Delta (n,k^{|\Delta (n,k,m\left)\right|},1)+\Delta (n,k,m),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \Phi (n,k)& =\Delta (n,k,k).\hfill \end{array}$$

(6)

For example, with $n=3$, $k=2$, and $m=1$, we have $\Phi (2,2)=\{e_0,e_1,2e_1\}$ by (3), so by (4),

$$\begin{array}{c}\hfill \Delta (3,2,1)=E_3(\Phi (2,2))=E_3\left(\{e_0,e_1,2e_1\}\right).\end{array}$$

There are three homotheties $h_1,h_2,h_3\in H$ that map $S_2=\{e_0,e_1\}$ into $\{e_0,e_1,2e_1\}$, defined for all $v\in {\mathbb{R}}^2$ by $h_1\left(v\right)=v,$$h_2\left(v\right)=2v,$ and $h_3\left(v\right)=e_1+v.$ To check this, recall that $e_0=(0,0)$. Hence $|\Delta (3,2,1\left)\right|\le 6$ since

$$\begin{array}{cccc}\hfill E_3\left(\{e_0,e_1,2e_1\}\right)& =\bigcup \{h\left(S_3\right):h\in H,h\left(S_2\right)\subseteq \{e_0,e_1,2e_1\}\}\hfill & \hfill & \mathrm{by}\left(2\right)\hfill \\ & =\bigcup \{h\left(S_3\right):h\in \{h_1,h_2,h_3\}\}\hfill \\ & =h_1\left(S_3\right)\cup h_2\left(S_3\right)\cup h_3\left(S_3\right)\hfill \\ & =h_1\left(\{e_0,e_1,e_2\}\right)\cup h_2\left(\{e_0,e_1,e_2\}\right)\cup h_3\left(\{e_0,e_1,e_2\}\right)\hfill & \hfill & \mathrm{by}\left(1\right)\hfill \\ & =\{e_0,e_1,e_2\}\cup \{e_0,2e_1,2e_2\}\cup \{e_1,2e_1,e_1+e_2\}\hfill \\ & =\{e_0,e_1,e_2,2e_1,2e_2,e_1+e_2\}.\hfill \end{array}$$

If $|\Delta (3,2,1\left)\right|=6$ then

$$\begin{array}{cccc}\hfill \Phi (3,2)& =\Delta (3,2,2)\hfill & \hfill & \mathrm{by}\left(6\right)\hfill \\ & =\Delta (3,2^{|\Delta (3,2,1\left)\right|},1)+\Delta (3,2,1)\hfill & \hfill & \mathrm{by}\left(5\right)\hfill \\ & =\Delta (3,2^6,1)+\{e_0,e_1,e_2,2e_2,e_1+e_2\}\hfill \\ & =E_3(\Phi (2,64))+\{e_0,e_1,e_2,2e_2,e_1+e_2\}\hfill & \hfill & \mathrm{by}\left(4\right)\hfill \end{array}$$

Since $\Phi (2,64)$ has 65 points by (3), there are $\left({\displaystyle \genfrac{}{}{0pt}{}{65}{2}}\right)=2080$ homotheties mapping $S_2$ into $\Phi (2,64)$, hence $E_3(\Phi (2,64))$ may contain as many as $65+2080=2145$ points, and $\Phi (3,2)$ may contain as many as $2145+6=2151$.

For integers $n\ge 3$, $k\ge 1$, and $m\ge 1$, let $A(n,k)$ be the statement that

for every k-coloring of $\Phi (n,k)$, $\Phi (n,k)$ contains a monochromatic homothetic image of $S_n$,

and let $D(n,k,m)$ be the statement that

for every k-coloring f of $\Delta (n,k,m)$ there are scalars ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_m\in \mathbb{R}$ and a vector $a\in {\mathbb{R}}^2$ such that

$$0={\lambda }_0<{\lambda }_1<\dots <{\lambda }_m$$

and if

$$h_{i,j}\left(v\right)=a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)v$$

whenever $v\in {\mathbb{R}}^2$ and $0\le i<j\le m$, then

$$h_{i,j}\left(S_n\right)\subseteq \Delta (n,k,m)$$

and

$$\left|f\right(h_{i,j}\left(S_{n-1}\right)\left)\right|=1.$$

Theorem 1. 

If $3\le n\in \omega$, $1\le k\in \omega$, and $1\le m\in \omega$, then $A(n,k)$ and $D(n,k,m)$.

• The proof is by a double induction. For each of the definitions (3)–(6), we show that if the sets on the right side have the requisite properties, then the set on left side does so as well. We will prove that the following statements hold whenever $n\ge 3$, $k\ge 1$, and $m\ge 1$.

1.

$A(2,k)$,

2.

if $A(n-1,k)$ then $D(n,k,1)$,

3.

if $D(n,k,m)$ and $D(n,k^{|\Delta (n,k,m\left)\right|},1)$ then $D(n,k,m+1)$,

4.

if $D(n,k,k)$ then $A(n,k)$.

To see that these statements are enough to establish the theorem, we lay out explicitly the initial inductive steps. By (1) we have

(a) $A(2,k)$ for all $k\ge 2$.

• For any $k\ge 2$ we get $D(3,k,1)$ by (a) and (2) with $n=3$. This proves

(b) $D(3,k,1)$ for all $k\ge 2$.

• For any particular $k\ge 2$, we have by (b) not only $D(3,k,1)$ but also $D(3,k^{|\Delta (3,k,1\left)\right|},1)$, hence $D(3,k,2)$ by (3) with $m=1$. This proves

(c) $D(3,k,2)$ for all $k\ge 2$.

• For any particular $k\ge 2$, we have $D(3,k,2)$ by (c) and also $D(3,k^{|\Delta (3,k,2\left)\right|},1)$ by (b), hence $D(3,k,3)$ by (3) with $m=2$. This proves

(d) $D(3,k,3)$ for all $k\ge 2$.

• The reasoning that leads from (c) to (d) can be repeated any finite number of times, leading (by induction) to the conclusion that

(e) $D(3,k,m)$ for all $k\ge 2$ and $m\ge 1$.

• As a set of particular cases of (e) we have $D(3,k,k)$ for all $k\ge 2$. It follows by (iv) that

(f) $A(3,k)$ for all $k\ge 2$.

• The deduction from (a) to (f) is just the first step (from $n=2$ to $n=3$) in an inductive proof that

(g) $A(n,k)$ for all $k\ge 2$ and all $n\ge 2$.

• Note that (g) is the usual statement of Gallai’s theorem, but the statement we have given above incorporates all the statements about $\Delta$ that were mentioned in the exposition of the double induction.

3. Proof of $A(2,k)$ for $k\ge 1$

Assume $k\ge 1$. Since $e_0=(0,0)$, definition (3) implies that

$$\begin{array}{c}\hfill \Phi (2,k)=\{e_0,e_1,2e_1,\dots ,k{e}_1\}.\end{array}$$

Assume that f is a k-coloring of $\Phi (2,k)$. We must show that $\Phi (2,k)$ contains a monochromatic homothetic image of $S_2=\{e_0,e_1\}$. From $e_0\ne e_1$ it follows that the number of points in $\Phi (2,k)$ is $k+1$, one more than the number of colors, so there are (at least) two points in $\Phi (2,k)$ that get the same color, say $f\left(i{e}_1\right)=f\left(j{e}_1\right)$ for some $i,j$ such that $0\le i<j\le k$. Define h by $h\left(v\right)=i{e}_1+(j-i)v$ for all $v\in {\mathbb{R}}^2$. Then $h\in H$ since $j-i>0$ and h maps $S_2=\{e_0,e_1\}$ to $\{i{e}_1,j{e}_1\}\subseteq \Phi (2,k)$ because

$$\begin{array}{cc}\hfill h\left(e_0\right)& =i{e}_1+(j-i)e_0=i{e}_1+(j-i)(0,0)=i{e}_1,\hfill \\ \hfill h\left(e_1\right)& =i{e}_1+(j-i)e_1=i{e}_1+j{e}_1-i{e}_1=j{e}_1.\hfill \end{array}$$

This homothetic image is monochromatic because $f\left(i{e}_1\right)=f\left(j{e}_1\right)$.

4. Proof of $D(n,k,1)$ from $n\ge 3$, $k\ge 1$, and $A(n-1,k)$

Assume $n\ge 3$, $k\ge 1$, and f is a k-coloring of $\Delta (n,k,1)$. By the inductive assumption $A(n-1,k)$, $\Phi (n-1,k)$ contains a monochromatic homothetic image of $S_{n-1}$, so there is some $h\in H$ such that

$$\begin{array}{c}\hfill h\left(S_{n-1}\right)\subseteq \Phi (n-1,k),\end{array}$$

(7)

$$\begin{array}{c}\hfill \left|f\right(h\left(S_{n-1}\right)\left)\right|=1.\end{array}$$

(8)

From (7), the definition of $E_n$, and definition (4), it follows that

$$h\left(S_n\right)\subseteq E_n(\Phi (n-1,k))=\Delta (n,k,1).$$

(9)

To conclude that $D(n,k,1)$, we must find ${\lambda }_0,{\lambda }_1\in \mathbb{R}$ and $a\in {\mathbb{R}}^2$ such that $0={\lambda }_0<{\lambda }_1$, $h_{0,1}\left(S_n\right)\subseteq \Delta (n,k,1)$, and $\left|f\right(h_{0,1}\left(S_{n-1}\right)\left)\right|=1$, where

$$h_{0,1}\left(v\right)=a+{\lambda }_0{e}_n+({\lambda }_1-{\lambda }_0)v=a+{\lambda }_{1}v$$

whenever $v\in {\mathbb{R}}^2$. It suffices to let ${\lambda }_1$ and a be the dilation scalar and displacement vector associated with h, for then $h_{0,1}=h$ and the two desired equations are (8) and (9).

5. Proof of $D(n,k,m+1)$ from $D(n,k,m)$ and $D(n,k^{|\Delta (n,k,m\left)\right|},1)$

Suppose f is a k-coloring of $\Delta (n,k,m+1)$. By definition (5), f assigns a color to every vector obtained by adding a vector from $\Delta (n,k^{|\Delta (n,k,m\left)\right|},1)$ to a vector in $\Delta (n,k,m)$. Therefore, for every $v\in \Delta (n,k^{|\Delta (n,k,m\left)\right|},1)$, we may let $f^{\prime }\left(v\right)$ be the k-coloring of $\Delta (n,k,m)$ defined by

$$f^{\prime }\left(v\right)\left(w\right)=f(v+w)\mathrm{for}\mathrm{every}w\in \Delta (n,k,m).$$

(10)

This gives us a new coloring $f^{\prime }$ that assigns each vector $v\in \Delta (n,k^{|\Delta (n,k,m\left)\right|},1)$ to an element $f^{\prime }\left(v\right)$ of the set of k-colorings of $|\Delta (n,k,m\left)\right|$, a set whose cardinality is $k^{|\Delta (n,k,m\left)\right|}$. Thus $f^{\prime }$ is a $k^{|\Delta (n,k,m\left)\right|}$-coloring of $\Delta (n,k^{|\Delta (n,k,m\left)\right|},1)$. From the inductive hypothesis $D(n,k^{|\Delta (n,k,m\left)\right|},1)$, applied to the coloring $f^{\prime }$, we know there are ${\mu }_0,{\mu }_1\in \mathbb{R}$ and $b\in {\mathbb{R}}^2$ such that $0={\mu }_0<{\mu }_1$ and, defining $g_{0,1}$ by

$$g_{0,1}\left(v\right)=b+{\mu }_0{e}_n+({\mu }_1-{\mu }_0)v=b+{\mu }_{1}v\mathrm{for}\mathrm{all}v\in {\mathbb{R}}^2,$$

(11)

we have

$$\begin{array}{cc}\hfill & g_{0,1}\left(S_n\right)\subseteq \Delta (n,k^{|\Delta (n,k,m\left)\right|},1),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & |f^{\prime }\left(g_{0,1}\left(S_{n-1}\right)\right)|=1.\hfill \end{array}$$

(13)

From the inductive hypothesis $D(n,k,m)$, applied to the k-coloring $f^{\prime }\left(b\right)$, we know there are ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_m\in \mathbb{R}$ and $a\in {\mathbb{R}}^2$ such that

$$0={\lambda }_0<{\lambda }_1<\dots <{\lambda }_m$$

(14)

and, assuming

$$h_{i,j}\left(v\right)=a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)v\mathrm{whenever}v\in {\mathbb{R}}^2\mathrm{and}0\le i<j\le m,$$

(15)

we have

$$\begin{array}{cc}\hfill & h_{i,j}\left(S_n\right)\subseteq \Delta (n,k,m),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & |f^{\prime }\left(b\right)\left(h_{i,j}\left(S_{n-1}\right)\right)|=1.\hfill \end{array}$$

(17)

Figure 1 illustrates this construction for the case $m=n=4$; the values are chosen only to make the recursive geometry visible in a nondegenerate example.

Next we prove that if $w\in \Delta (n,k,m)$, $0\le \ell <n-1$, and $0\le i<m$, then

$$\begin{array}{cc}\hfill f(b+w)& =f(b+{\mu }_1{e}_{\ell }+w),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill f(b+a+{\lambda }_i{e}_n)& =f(b+a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_{\ell }).\hfill \end{array}$$

(19)

Proof of (18): If $0\le \ell <n-1$, then $e_0,e_{\ell }\in S_{n-1}$ and, by (11), $g_{0,1}\left(e_0\right)=b$ and $g_{0,1}\left(e_{\ell }\right)=b+{\mu }_1{e}_{\ell }$, hence $b,b+{\mu }_1{e}_{\ell }\in g_{0,1}\left(S_{n-1}\right)$ so from (13) we conclude that $f^{\prime }\left(b\right)=f^{\prime }(b+{\mu }_1{e}_{\ell })$. This holds whenever $0\le l<n-1$, so by (10), we obtain (18).

Proof of (19): If $0\le \ell <n-1$ and $0\le i<m$ then $e_0,e_{\ell }\in S_{n-1}$ and, by (15) applied with $0\le i<j=m$, we have

$$\begin{array}{cc}\hfill a+{\lambda }_i{e}_n& =h_{i,m}\left(e_0\right)\in h_{i,m}\left(S_{n-1}\right),\hfill \\ \hfill a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_{\ell }& =h_{i,m}\left(e_{\ell }\right)\in h_{i,m}\left(S_{n-1}\right).\hfill \end{array}$$

These two vectors are in $\Delta (n,k,m)$ by (16) and get the same color from $f^{\prime }\left(b\right)$ by (17) with $j=m$, i.e.,

$$f^{\prime }\left(b\right)(a+{\lambda }_i{e}_n)=f^{\prime }\left(b\right)(a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_{\ell }).$$

Now (19) follows from this equation by (10) with $v=b$.

Define a new system $h^{\prime }$ as follows. Let ${\lambda }_{m+1}={\mu }_1+{\lambda }_m$ and let

$$h_{i,j}^{\prime }\left(v\right)=b+a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)v\mathrm{if}v\in {\mathbb{R}}^2\mathrm{and}0\le i<j\le m+1.$$

(20)

To conclude that $D(n,k,m+1)$, we must now show for this new system that

$$0={\lambda }_0<{\lambda }_1<\dots <{\lambda }_m<{\lambda }_{m+1},$$

which we get immediately from (14), $0<{\mu }_1$, and ${\lambda }_m<{\mu }_1+{\lambda }_m={\lambda }_{m+1}$, and, assuming $v\in {\mathbb{R}}^2$ and $0\le i<j\le m+1$, that

$$\begin{array}{cc}\hfill & h_{i,j}^{\prime }\left(S_n\right)\subseteq \Delta (n,k,m+1),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \left|f\right(h_{i,j}^{\prime }\left(S_{n-1}\right)\left)\right|=1.\hfill \end{array}$$

(22)

Case 1. $0\le i<j\le m$.

$$\begin{array}{cccc}\hfill h_{i,j}^{\prime }\left(S_n\right)& =b+h_{i,j}\left(S_n\right)\hfill & \hfill & \mathrm{by}\left(15),\right(20)\hfill \\ & \subseteq b+\Delta (n,k,m)\hfill & \hfill & \mathrm{by}\left(16\right)\hfill \\ & =g_{0,1}\left(e_0\right)+\Delta (n,k,m)\hfill & \hfill & \mathrm{by}(11),e_0=(0,0)\hfill \\ & \subseteq g_{0,1}\left(S_n\right)+\Delta (n,k,m)\hfill & \hfill & e_0\in S_n\hfill \\ & \subseteq \Delta (n,k^{|\Delta (n,k,m\left)\right|},1)+\Delta (n,k,m)\hfill & \hfill & \mathrm{by}\left(12\right)\hfill \\ & =\Delta (n,k,m+1)\hfill & \hfill & \mathrm{by}\left(5\right)\hfill \end{array}$$

so (21) holds in this case, and

$$\begin{array}{cccc}\hfill \left|f\right(h_{i,j}^{\prime }\left(S_{n-1}\right)\left)\right|& =\left|f\right(b+h_{i,j}\left(S_{n-1}\right)\left)\right|\hfill & \hfill & \mathrm{by}\left(15),\right(20)\hfill \\ & =|f^{\prime }\left(b\right)\left(h_{i,j}\left(S_{n-1}\right)\right)|\hfill & \hfill & \mathrm{by}\left(10\right)\hfill \\ & =1\hfill & \hfill & \mathrm{by}\left(17\right)\hfill \end{array}$$

so (22) also holds in this case.

Case 2. $0\le i<j=m+1$. Proof of (21): If $v\in S_n$, then

$$\begin{array}{cccc}\hfill h_{i,m+1}^{\prime }\left(v\right)& =b+a+{\lambda }_i{e}_n+({\lambda }_{m+1}-{\lambda }_i)v\hfill & \hfill & \left(20\right)\hfill \\ & =b+a+{\lambda }_i{e}_n+({\lambda }_m+{\mu }_1-{\lambda }_i)v\hfill & \hfill & \mathrm{def}{\lambda }_{m+1}\hfill \\ & =b+{\mu }_{1}v+a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)v\hfill \\ & =g_{0,1}\left(v\right)+h_{i,m}\left(v\right)\hfill & \hfill & \left(11),\right(15)\hfill \\ & \in \Delta (n,k^{|\Delta (n,k,m\left)\right|},1)+\Delta (n,k,m)\hfill & \hfill & \left(12),\right(16)\hfill \\ & =\Delta (n,k,m+1)\hfill & \hfill & \left(5\right)\hfill \end{array}$$

For (22), consider an arbitrary ℓ with $0\le \ell <n-1$, i.e., any arbitrary $e_{\ell }\in S_{n-1}$. Note that by (15) and (16) with $j=m$ we have

$$a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_{\ell }=h_{i,m}\left(e_{\ell }\right)\in \Delta (n,k,m).$$

(23)

Hence, noting that $i<m+1$,

$$\begin{array}{cccc}\hfill f\left(h_{i,m+1}^{\prime }\left(e_0\right)\right)& =f(b+a+{\lambda }_i{e}_n)\hfill & \hfill & \mathrm{by}(20),e_0=(0,0)\hfill \\ & =f(b+a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_l)\hfill & \hfill & \mathrm{by}\left(19\right)\mathrm{if}i<m,\mathrm{trivial}\mathrm{if}i=m\hfill \\ & =f(b+{\mu }_1{e}_l+a+{\lambda }_i{e}_n+({\lambda }_m-{\lambda }_i)e_l)\hfill & \hfill & \mathrm{by}\left(23),\right(18)\hfill \\ & =f(b+a+{\lambda }_i{e}_n+({\mu }_1+{\lambda }_m-{\lambda }_i)e_l)\hfill & \hfill & \mathrm{by}\mathrm{computation}\hfill \\ & =f(b+a+{\lambda }_i{e}_n+({\lambda }_{m+1}-{\lambda }_i)e_l)\hfill & \hfill & \mathrm{by}\mathrm{definition}\mathrm{of}{\lambda }_{m+1}\hfill \\ & =f\left(h_{i,m+1}^{\prime }\left(e_l\right)\right)\hfill & \hfill & \mathrm{by}\left(20\right)\hfill \end{array}$$

Therefore, (22) holds in this case.

6. Proof of $A(n,k)$ from $D(n,k,k)$

Since $\Phi (n,k)=\Delta (n,k,k)$, we need only show that $\Delta (n,k,k)$ contains a monochromatic homothetic image of $S_n$, assuming f is a k-coloring of $\Delta (n,k,k)$. From the inductive hypothesis $D(n,k,k)$, we know there are ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_k\in \mathbb{R}$ and $a\in {\mathbb{R}}^2$ such that $0={\lambda }_0<{\lambda }_1<\dots <{\lambda }_k$ and if

$$h_{i,j}\left(v\right)=a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)v$$

whenever $v\in {\mathbb{R}}^2$ and $0\le i<j\le k$, then $h_{i,j}\left(S_n\right)\subseteq \Delta (n,k,k)$ and $\left|f\right(h_{i,j}\left(S_{n-1}\right)\left)\right|=1$. Let’s look at the images of $e_n$ and $e_0$ under all these homotheties $h_{i,j}$ with $0\le i<j\le k$. We have

$$h_{i,j}\left(e_n\right)=a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)e_n=a+{\lambda }_j{e}_n,$$

and, since $e_0=(0,0)$,

$$h_{i,j}\left(e_0\right)=a+{\lambda }_i{e}_n+({\lambda }_j-{\lambda }_i)e_0=a+{\lambda }_i{e}_n.$$

The set of images of $e_n$ is

$$\{a+{\lambda }_j{e}_n:0<j\le k\}$$

The set of images of $e_0$ is

$$\{a+{\lambda }_i{e}_n:0\le i<k\}$$

The union of these two sets, namely

$$\{a+{\lambda }_{\ell }{e}_n:0\le \ell \le k\}$$

is a $(k+1)$-element set, to which are assigned only k colors. Two elements receive the same color, hence there are $i,j$ such that $0\le i<j\le k$ and

$$f(a+{\lambda }_i{e}_n)=f(a+{\lambda }_j{e}_n).$$

However, $a+{\lambda }_i{e}_n=h_{i,j}\left(e_0\right)$ and $a+{\lambda }_j{e}_n=h_{i,j}\left(e_n\right)$, so

$$f\left(h_{i,j}\left(e_0\right)\right)=f\left(h_{i,j}\left(e_n\right)\right).$$

Of course, we already know $\left|f\right(h_{i,j}\left(S_{n-1}\right)\left)\right|=1$, so this last equation tells us, since $e_0\in S_{n-1}$, that $e_n$ also receives the same color as all the other elements of $S_{n-1}$. But $S_n=S_{n-1}\cup \left\{e_n\right\}$, so in fact we have shown $\left|f\right(h_{i,j}\left(S_n\right)\left)\right|=1$. We also know that $h_{i,j}\left(S_n\right)\subseteq \Delta (n,k,k)$, so we conclude that $\Delta (n,k,k)$ does indeed contain a monochromatic homothetic image of $S_n$.

7. Concluding Remarks

Gallai’s theorem, a central result in Euclidean Ramsey theory, can be simply deduced as a consequence of the Hales-Jewett theorem, but the purpose here has been a detailed analysis of Witt’s proof, assuming nothing more than the properties of an ordered ring. The key steps and constructions have been isolated in the recursive definitions of the sets $\Phi (n,k)$ and $\Delta (n,k,m)$ and the statements $A(n,k)$ and $D(n,k,m)$. These may make Gallai’s theorem more accessible and could form the basis of further formalization, generalization, or upper-bound estimates for the numbers of points required for monochromatic copies to occur.