Further improving of upper bound on a geometric Ramsey problem

TL;DR

This paper refines the upper bound on the dimension needed to guarantee a monochromatic coplanar 4-vertex clique in any 2-coloring of the hypercube's edges, improving previous bounds significantly.

Contribution

It improves the known upper bound on the geometric Ramsey problem from a triple exponential tower to a lower triple exponential tower.

Findings

Reduced the upper bound from 2↑↑↑6 to 2↑↑↑5.

Provided a tighter estimate for the minimal dimension n.

Revisited and improved upon previous combinatorial bounds.

Abstract

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed by Graham and Rothschild and they gave an upper bound: $n\le F(F(F(F(F(F(F(12)))))))$, where $F(m) = 2\uparrow^m3$. In 2014 Lavrov, Lee and Mackey greatly improved this result by giving upper bound $n< 2\uparrow\uparrow\uparrow 6 < F(5)$. In this paper we revisit their estimates and reduce upper bound to $n< 2\uparrow\uparrow\uparrow 5$

Equations

$${\forall }_{1\le i\le k}\left[{\exists }_{1\le j\le d}{\rho }_i\left(y_1,\dots ,y_d\right)=y_d\vee {\exists }_{0\le j\le n-1}{\rho }_i\left(y_1,\dots ,y_d\right)=j\right],$$

$${\forall }_{1\le i\le k}\left[{\exists }_{1\le j\le d}{\rho }_i\left(y_1,\dots ,y_d\right)=y_d\vee {\exists }_{1\le j\le d}{\rho }_i\left(y_1,\dots ,y_d\right)=n-1-y_d\vee {\exists }_{0\le j\le n-1}{\rho }_i\left(y_1,\dots ,y_d\right)=j\right],$$

$${\forall }_{1\le i\le k}\left[{\exists }_{1\le j\le d}{\rho }_i\left(y_1,\dots ,y_d\right)=y_d\vee {\exists }_{1\le j\le d}{\rho }_i\left(y_1,\dots ,y_d\right)=-y_d\vee {\exists }_{0\le j\le n-1}{\rho }_i\left(y_1,\dots ,y_d\right)=j\right],$$

$${\forall }_{1\le i\le k}\left[{\exists }_{1\le j\le l}{\pi }_i\left(y_1,\dots ,y_l\right)=y_j\vee {\exists }_{0\le j\le n-1}{\pi }_i\left(y_1,\dots ,y_l\right)=j\right],$$

$${\forall }_{y\in Y}D\left(\pi \left(y\right)\right)=D^{\prime }\left(y\right),$$

$${\forall }_{1\le i\le l}{D}^{\prime }\left(y_1,y_2,\dots ,y_{i-1},n-1,y_{i+1},\dots ,y_l\right)=D^{\prime }\left(y_1,y_2,\dots ,y_{i-1},n-2,y_{i+1},\dots ,y_l\right).$$

$$f\left(l,k\right)=\{\begin{array}{ll}{k}^{f{\left(l-1,k\right)}^{2l-2}}+1& \text{for }l>1\\ k+1& \text{for }l=1\end{array}.$$

$$f\left(l,k\right)<k^{{\left(k\uparrow \uparrow 2l-2\right)}^{2l-2}}<k^{{\left(k\uparrow \uparrow 2l-2\right)}^k}<k\uparrow \uparrow 2l.$$

$$\text{TTT}\left(n+2,c,d\right)\le \text{Cub}\left(n+2,c,\text{Cub}\left(n+1,c,\text{TTT}\left(n,c,d\right)\right)\right).$$

$$X$$

$$Y$$

$$Y^{\prime }$$

$$Z$$

$$Z^{\prime }$$

$${\rho }^{\prime }\left(x_1,x_2,\dots ,x_d\right)=\pi \circ \gamma \circ \sigma \circ \tau \circ \zeta \circ \rho \left(x_1,x_2,\dots ,x_d\right).$$

$${\forall }_{1\le i\le \dim X}\left[{\exists }_{1\le j\le d}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=x_j+1\vee {\exists }_{1\le j\le d}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=n-x_j\vee {\exists }_{1\le j\le n}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=j\right].$$

$${\rho }_i^{\prime \prime }\left(x_1,\dots ,x_d\right)=\{\begin{array}{ll}{x}_j& \text{if }{\exists }_{1\le j\le d}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=x_j+1\\ n+1-x_j& \text{if }{\exists }_{1\le j\le d}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=n-x_j\\ j& \text{if }{\exists }_{1\le j\le n}{\rho }_i^{\prime }\left(x_1,\dots ,x_d\right)=j\end{array}.$$

$$\text{Cub}\left(3,2,\text{TTT}\left(2,2,6\right)\right)$$

$$\le 428\cdot f\left(428,2^{3^{428}}\right)\le 428\cdot \left(2^{3^{428}}\uparrow \uparrow 856\right)$$

$$<2^9\cdot \left(\left(2\uparrow \uparrow 6\right)\uparrow \uparrow 856\right)<2\uparrow \uparrow \left(6\cdot 856+1\right)=2\uparrow \uparrow 5137.$$

$$\text{TTT}\left(4,2,6\right)+1$$

$$\le 1+\text{Cub}\left(4,2,2\uparrow \uparrow 5137\right)$$

$$\le 1+\left(2\uparrow \uparrow 5137\right)\cdot f\left(2\uparrow \uparrow 5137,2^{4^{2\uparrow \uparrow 5137}}\right)$$

$$\le 1+\left(2\uparrow \uparrow 5137\right)\cdot \left(\left(2^{4^{2\uparrow \uparrow 5137}}\right)\uparrow \uparrow \left(2\cdot 2\uparrow \uparrow 5137\right)\right)$$

$$<\left(2\uparrow \uparrow 5138\right)\cdot \left(\left(2\uparrow \uparrow 5140\right)\uparrow \uparrow \left(2\cdot 2\uparrow \uparrow 5137\right)\right)$$

$$<\left(2\uparrow \uparrow 5138\right)\cdot \left(2\uparrow \uparrow \left(5140\cdot 2\cdot 2\uparrow \uparrow 5137\right)\right)$$

$$<2\uparrow \uparrow \left(10280\cdot 2\uparrow \uparrow 5137+1\right)$$

$$<2\uparrow \uparrow \left(2\uparrow \uparrow 5138\right).$$

Full text

1 Setting

Definition 1**.**

Given $n,c,d\in\mathbb{Z}_{+}$ let Hales-Jewett number $\text{HJ}\left(n,c,d\right)$ be the least integer $k$ with the following property. For any $c$-coloring $D$ of $\left\{0,1,\ldots,n-1\right\}^{k}$ there exists an injective function $\rho:\left\{0,1,\ldots,n-1\right\}^{d}\rightarrow\left\{0,1,\ldots,n-1\right\}^{k}$ such, that

[TABLE]

and $\rho\left(\left\{0,1,\ldots,n-1\right\}^{d}\right)$ is $D$-monochromatic.

Definition 2**.**

Given $n,c,d\in\mathbb{Z}_{+}$ let Tic-Tac-Toe number $\text{TTT}\left(n,c,d\right)$ be the least integer $k$ with the following property. For any $c$-coloring $D$ of $\left\{0,1,\ldots,n-1\right\}^{k}$ there exists an injective function $\rho:\left\{0,1,\ldots,n-1\right\}^{d}\rightarrow\left\{0,1,\ldots,n-1\right\}^{k}$ such, that

[TABLE]

and $\rho\left(\left\{0,1,\ldots,n-1\right\}^{d}\right)$ is $D$-monochromatic. Image of such a function is called a $d$-dimensional Tic-Tac-Toe Subspace.

Definition 3**.**

Given $d\in\mathbb{Z}_{+}$ let $\text{Graham}\left(d\right)$ be the smallest dimension $k$ such that for every edge-coloring of a complete graph on the points $\{\pm 1\}^{k}$ there exists an injective function $\rho:\left\{\pm 1\right\}^{d}\rightarrow\left\{\pm 1\right\}^{k}$ with

[TABLE]

and all edges between the points of $\rho\left(\left\{\pm 1\right\}^{d}\right)$ have the same color.

In particular, $\text{Graham}\left(2\right)$ is the smallest integer $k$, such that for every edge-coloring of a complete graph on the points $\{\pm 1\}^{k}$ there exist four coplanar vertices such that all six edges between them are monochromatic. Our goal is to give a better upper bound for that value. It has been proven in [2], that $\text{Graham}\left(2\right)\leq\text{TTT}\left(4,2,6\right)+1$, and then, using obvious inequality $\text{HJ}\left(n,c,d\right)\geq\text{TTT}\left(n,c,d\right)$ it was shown that $\text{Graham}\left(2\right)<2\uparrow\uparrow\uparrow 6$. Our approach is to not use the Hales-Jewett function, because TTT$\left(\cdot,c,d\right)$ and HJ$\left(\cdot,c,d\right)$ have similar growth rate, but initial values of TTT$\left(\cdot,c,d\right)$ are much smaller.

Definition 4**.**

Given $n,c,l\in\mathbb{Z}_{+}$ let $\text{Cub}\left(n,c,l\right)$ be the least integer $k$ with the following property. For any $c$-coloring $D$ of $X=\left\{0,1,\ldots,n-1\right\}^{k}$ there exists $c$-coloring $D^{\prime}$ of $Y=\left\{0,1,\ldots,n-1\right\}^{l}$ and an injective function $\pi:Y\rightarrow X$ such, that

[TABLE]

[TABLE]

[TABLE]

In other words, values $n-2$ and $n-1$ are not distinguished by induced coloring of $Y$.

Lemma 1**.**

Let $n,c,l\in\mathbb{Z}_{+}$, then $\text{Cub}\left(n,c,l\right)\leq l\cdot f\left(l,c^{n^{l}}\right)$, where

[TABLE]

Proof.

This is straightforward conclusion from chapter 1 of [3]. This fact was used to show, that $\text{HJ}\left(n+1,c,d\right)\leq\text{Cub}\left(n+1,c,\text{HJ}\left(n,c,d\right)\right)\leq\text{HJ}\left(n,c,d\right)\cdot f\left(\text{HJ}\left(n,c,d\right),c^{{n+1}^{\text{HJ}\left(n,c,d\right)}}\right)$. ∎

Lemma 2**.**

Let $k,l\in\mathbb{Z}_{+}$ and $f$ be defined as above, then $2l<k\Rightarrow f\left(l,k\right)<k\uparrow\uparrow 2l.$

Proof.

For $l=1$ it is obviously true as $k+1<k^{k}=k\uparrow\uparrow 2$ for $k>2l=2$. By induction it is true for any $l$ because

[TABLE]

∎

Lemma 3**.**

Let $c,d\in\mathbb{Z}_{+}$, then $\text{TTT}\left(2,c,d\right)\leq\frac{c}{2}\cdot 3^{d}$.

Proof.

First, we notice $\text{TTT}\left(2,c,1\right)=\left\lceil\log_{2}\left(c+1\right)\right\rceil\leq c$ as a line connecting any two points in $\left\{0,1\right\}^{k}$ has property (1), so we just need to have more points than colors. Define $r_{1}=\left\lceil\log_{2}\left(c+1\right)\right\rceil,r_{i}=\left\lceil\log_{2}\left(c\cdot\prod_{j<i}\genfrac{(}{)}{0.0pt}{}{2^{r_{j}}}{2}+1\right)\right\rceil$, then by pigeonhole principle $\text{TTT}\left(2,c,d\right)\leq\sum_{j\leq d}r_{j}$. Because $r_{i}\leq 3\cdot r_{i-1}$ then $\sum_{j\leq d}r_{j}\leq r_{1}\cdot\frac{3^{d}-1}{2}$. ∎

Corollary 1**.**

By carefully repeating previous proof we can get even better estimate for certain values, in particular for $c=2$ we have $\left(r_{i}\right)=\left(2,4,11,32,95,284,\ldots\right)$ so $\text{TTT}\left(2,2,6\right)\leq 428$.

2 Main Result

Lemma 4**.**

For $n\geq 2$

[TABLE]

Proof.

We will basicaly repeat proof of Lemma 1.4 from [3], but with TTT instead of Hales-Jewett function. Define

[TABLE]

Let $D$ be any $c$-coloring of $X$, by definition of Cub we have induced $c$-coloring $D^{\prime}$ of $Y$ and embedding $\pi:Y\rightarrow X$ such that properties (2),(3),(4) hold (for $n:=n+2$). Again, by definition of Cub we have induced $c$-coloring $D^{\prime\prime}$ of $Z$ and embedding $\tau:Z\rightarrow Y^{\prime}$ such that properties (2),(3),(4) hold (for $n:=n+1$). By definition of TTT there exists an injective function $\rho:\left\{0,1,\ldots,n-1\right\}^{d}\rightarrow Z^{\prime}$ with property (1) and its image is $D^{\prime\prime}$-monochromatic. Let $\sigma:Y^{\prime}\rightarrow Y^{\prime}$ be defined as $\sigma\left(y_{1},y_{2},\ldots\right)=\left(n-y_{1},n-y_{2},\ldots\right)$, also let $\gamma:Y^{\prime}\rightarrow Y,\zeta:Z^{\prime}\rightarrow Z$ be natural embeddings.

Define $\rho^{\prime}:\left\{0,1,\ldots,n-1\right\}^{d}\rightarrow X$ as

[TABLE]

It is easy to check, that $\rho^{\prime}\left(\left\{0,1,\ldots,n-1\right\}^{d}\right)\subset\left\{1,2,\ldots,n\right\}^{\dim X}$ is $D$-monochromatic and

[TABLE]

Now, we define function $\rho^{\prime\prime}:\left\{0,1,\ldots,n,n+1\right\}^{d}\rightarrow X$ in a following way

[TABLE]

This function satisfies $\rho^{\prime}\left(x_{1},\ldots,x_{d}\right)=\rho^{\prime\prime}\left(x_{1}+1,\ldots,x_{d}+1\right)$, so $\rho^{\prime}\left(\left\{0,1,\ldots,n-1\right\}^{d}\right)=\rho^{\prime\prime}\left(\left\{1,2\ldots,n\right\}^{d}\right)$, and image of $\rho^{\prime\prime}$ is a Tic-Tac-Toe subspace. Because $\pi$ and $\tau$ have property (4) this image is also $D$-monochromatic, so $\text{TTT}\left(n+2,c,d\right)\leq\dim X.$ ∎

Lemma 5**.**

$\text{Cub}\left(3,2,\text{TTT}\left(2,2,6\right)\right)<2\uparrow\uparrow 5137$.

Proof.

[TABLE]

∎

Lemma 6**.**

$\text{TTT}\left(4,2,6\right)+1<2\uparrow\uparrow\left(2\uparrow\uparrow 5138\right)$.

Proof.

[TABLE]

∎

Theorem 1**.**

$\text{Graham}\left(2\right)<2\uparrow\uparrow\uparrow 5$.

Proof.

From [2] we know, that $\text{Graham}\left(2\right)\leq\text{TTT}\left(4,2,6\right)+1$, so $\text{Graham}\left(2\right)<2\uparrow\uparrow\left(2\uparrow\uparrow 5138\right)<2\uparrow\uparrow\uparrow 5$. ∎