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According to the Wikipedia article on Graham's number, the number found in the Graham and Rothschild paper "Ramsey's Theorem for n-Parameter Sets" (which has been called the "Little Graham's Number") is the best known upper bound to the Ramsey problem to which Graham's number is so famously associated.  This is not quite right.  Ever since Shelah published his paper "Primitive Recursive Bounds for Van der Waarden Numbers",  much lower bounds for the Graham's number problem, as well as many other Ramsey problems, have been obtainable.  However, as far as I know no one has bothered to obtain a precise upper bound on the Graham's number problem using Shelah's new methods.  That precise upper bound is the purpose of this article.

When I originally found an upper bound a few years ago, I used Spencer's proof of the Graham-Rothschild theorem.  I have been unable to dig up this proof however, so instead I will rely on the proof of the Graham-Rothschild theorem given in Shelah's paper.  This is enough to give a much lower upper bound.

I will not talk about Shelah's arguments proving the Graham-Rothschild theorem and related theorems, which are ingenious.  For that please download Shelah's paper.  I will instead simply state the inequalities generated by Shelah's arguments, and prove the upper bound based on those inequalities.

The first function we will have to bound is called simply \(f\).  I won't give the complicated definition;  all we need are the bounds

1) \(f(1, c) = c + 1\)

2) for \(n \ge 1, f(n+1, c) \le c^{f(n, c)^{2n}} + 1 \)

Lemma 1. \( a * b^c < b^{c + a} \) when \(b \ge 2\)

Proof. \(a * b^c < b^a * b^c = b^{c + a} \)

Lemma 2. \(b^c + a < b^{c + a} \) when \(a,c \ge 1, b \ge 2\)

Proof. \(b^c + a < b^c(1 + a) \le b^c * b^a = b^{c+a} \)

Proposition 3. For \(n, c \ge 2\), \( f(n, c) < c \uparrow c \uparrow \ldots \uparrow c \uparrow n^2 \) with \(n\) \(c\)'s.

Proof. We use induction.

\(n = 2: f(2, c) \le c^{{f(1, c)}^{2*1}} = c^{{c+1}^2} + 1 \le c^{c^4} = c^{c^{n^2}}\).

Assume the statement is true for f(n, c). Then

\( f(n+1, c) \le c^{{f(n, c)}^{2n}} + 1 \le c \uparrow (c \uparrow c \uparrow \ldots \uparrow c \uparrow n^2) \uparrow n\)

\(= c \uparrow (n * c \uparrow c \uparrow \ldots \uparrow c \uparrow n^2) \)

\(\le c \uparrow c \uparrow (n + c \uparrow c \uparrow \ldots \uparrow c \uparrow n^2) \) (using Lemma 1)

...

\( \le c \uparrow c \uparrow \ldots \uparrow c \uparrow (n^2 + n) \) (using multiple applications of Lemma 2)

\( < c \uparrow c \uparrow \ldots \uparrow c \uparrow (n+1)^2 \) with \( n+1\) \( c\)'s.

Our next function is a consequence of the Hales-Jewett Theorem.

Hales-Jewett Theorem.  For all \(n, c \ge 1 \), there exists a number \(HJ (n, c)\) such that whenever \(k \ge HJ(n, c) \) and the points \(\lbrace1, 2, \ldots, n\rbrace^k\) are colored with \(c\) colors, there exists a line of \(n\) points such that every point is the same color.

This is considered a major result in Ramsey Theory.  According to Graham, Rothschild, and Spencer in "Ramsey Theory", "Without this result (the Hales-Jewett Theorem), Ramsey Theory would more properly be called Ramseyian theorems."  Notice the similarity with the Graham's Number problem, except "line" is replaced by "point" and "plane" is replaced by "line".

Anyway... here are the bounds for \(HJ (n, c)\):

1) \( HJ (1, c) = 1\)

2) \( HJ (n+1, c) \le HJ(n, c) * f(HJ (n, c), c ^ {(n+1) ^ {HJ(n, c)}} ) \)

Lemma 4.  \( (a \uparrow \uparrow b) \uparrow \uparrow c \le a \uparrow \uparrow (b + 2c - 2) \)

Proof.  We use induction.

c = 1: \( (a \uparrow \uparrow b) \uparrow \uparrow 1 = a \uparrow \uparrow b = a \uparrow \uparrow (b + 2c - 2) \).

Now assume the statement for c.  Then

\( (a \uparrow \uparrow b) \uparrow \uparrow (c+1) = (a \uparrow \uparrow b) ^ {(a \uparrow \uparrow b) \uparrow \uparrow c} \) \(\le (a \uparrow \uparrow b) ^ {a \uparrow \uparrow (b + 2c - 2)} = (a ^ {a \uparrow \uparrow b-1})^{a \uparrow \uparrow (b + 2c - 2)} \) \(= a ^ {(a \uparrow \uparrow b-1) (a \uparrow \uparrow (b + 2c - 2) )} \le a ^ {(a \uparrow \uparrow (b + 2c - 2) )^2 } \) \(\le a ^ {(a \uparrow \uparrow (b + 2c - 1))} = a \uparrow \uparrow (b + 2c) = a \uparrow \uparrow (b + 2(c+1) - 2)\).

Proposition 5. For \( n, c \ge 2, HJ(n, c) < c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) \) with \(n\) terms in the series.

Proof.  We use induction on n.

n = 2: \(HJ(2, c) \le HJ(1, c) * f(HJ (1, c), c ^ {(1+1) ^ {HJ(1, c)}} ) = 1 * f(1, c^{2^1}) = f(1, c^2) = c^2 + 1 < c^{c^c} \le c \uparrow \uparrow (c + 1) \).

Now assume the statement for a given n.

\( HJ (n+1, c) \le HJ(n, c) * f(HJ (n, c), c ^ {(n+1) ^ {HJ(n, c)}} ) <  HJ(n, c) * f(HJ (n, c), HJ(n, c) ^ {HJ(n, c) ^ {HJ(n, c)}} ) \) \(< HJ (n, c) * (HJ(n, c) ^ {HJ(n, c) ^ {HJ(n, c)}}) \uparrow (HJ(n, c) ^ {HJ(n, c) ^ {HJ(n, c)}}) \uparrow \ldots \uparrow (HJ(n, c) ^ {HJ(n, c) ^ {HJ(n, c)}}) \uparrow (HJ(n, c)^2) \) (using Proposition 3)

\( < (HJ(n, c) ^ {HJ(n, c) ^ {HJ(n, c)}}) \uparrow \uparrow (HJ(n, c) + 1) = (HJ(n,c) \uparrow \uparrow 3) \uparrow \uparrow (HJ(n, c) + 1) \le HJ(n, c) \uparrow \uparrow (3 + 2 HJ(n, c) ) \) (using Lemma 4)

\( < (c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) ) \uparrow \uparrow (3 + 2 (c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) ) )\)

\(\le c \uparrow \uparrow (c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) + 4(c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) ) + 4) \)

\(<  c \uparrow \uparrow (c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n) )\)

\(=  c \uparrow \uparrow c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n ) \) with n+1 terms in the series.

Corollary 6.  If \(n \le c, HJ(n, c) < c \uparrow \uparrow \uparrow (n+1) \)

Proof. \(HJ(n, c) < c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c + n - 1) < c \uparrow \uparrow c \uparrow \uparrow \ldots c \uparrow \uparrow (c \uparrow \uparrow c) = c \uparrow \uparrow \uparrow (n+1) \)

Next, we visit the classical Ramsey Theorem for Hypergraphs.

Ramsey Theorem for Hypergraphs.  For any \(t, l, c \ge 1 \), there exists a number \(RAM(t, l, c)\) such that whenever \(m \ge RAM(t, l, c)\) and the subsets of size \(l\) of \(\lbrace1, 2, \ldots, m \rbrace \) are colored with \( c\) colors, there exists a subset \(S\)of size \(t\) such that all subsets of \(S\) of size \(l\) are colored the same color.

Now we can go to our main theorem, which is the Graham-Rothschild Theorem for n-paramenter subsets.

Graham-Rothschild Theorem.  For all \(n, t, l, c \ge 1 \) there exists a natural number \(GR(n, t, l, c) \) such that whenever \(k \ge GR(n, t, l, c) \) and the \(l\)-parameter subsets of \(\lbrace1, 2, \ldots, n\rbrace^k\) are colored with \(c\) colors, there eixsts a \(t\)-parameter subset \(S\) such that all \(l\)-parameter subsets of \(S\) have the same color.

The Graham's Number problem is a special case of the Graham-Rothschild theorem for n-parameter subsets;  in particular, the bound we seek is GR(2, 2, 1, 2).

GR(n,t,l,c) is bounded as follows:

Let \(m = RAM (t, l, c) \).

Define \(k_i\) for \(i \le m\) by

\( k_0 = 0 \)

\(k_{i+1} = k_i + HJ(k_i + n, c ^ {(n + l) ^ {k_i + m - i - 1}}) \)

Then \(GR (n, t, l, c) \le k_m \).

So, let us bound GR(2, 2, 1, 2).  We set \(m = RAM (2, 1, 2) \), which is the smallest number of points such that, if we color the points with two colors, there is a set of two points with the same color.  Obviously \(m = 3 \).

Lemma 7.  \( (a \uparrow \uparrow b) \uparrow \uparrow \uparrow c \le a \uparrow \uparrow a \uparrow \uparrow \ldots a \uparrow \uparrow (b + c - 1) \) with \(c\) \( a\)'s.

Proof.  We use induction on c, of course.

c = 1:  \((a \uparrow \uparrow b) \uparrow \uparrow \uparrow 1 = a \uparrow \uparrow b = a \uparrow \uparrow (b + c - 1) \).

Now assume the statement for c.

\( (a \uparrow \uparrow b) \uparrow \uparrow \uparrow (c + 1) = (a \uparrow \uparrow b) \uparrow \uparrow ((a \uparrow \uparrow b) \uparrow \uparrow \uparrow c) \)

\( \le (a \uparrow \uparrow b) \uparrow \uparrow (a \uparrow \uparrow a \uparrow \uparrow \ldots a \uparrow \uparrow (b + c - 1) ) \)

\( < a \uparrow \uparrow (b - 2 + 2 a \uparrow \uparrow a \uparrow \uparrow \ldots a \uparrow \uparrow (b + c - 1) )\)

\(< a \uparrow \uparrow a \uparrow \uparrow a \uparrow \uparrow \ldots a \uparrow \uparrow (b + c) \) with \(c+1\) \( a\)'s.

We have

\( k_0 = 0 \)

\( k_1 =  0 + HJ(0 + 2, 2 ^ {(2 + 1) ^ {0 + 3 - 0 - 1}}) = HJ(2, 2^9) = 2^{18} + 1 \)

\(k_2 =  k_1 + HJ(k_1 + 2, 2 ^ {(2+1) ^ {k_1 + 3 - 1 - 1}})  = 2^{18} + 1 + HJ(2^{18} + 3, 2 ^ {3 ^ {(2^{18} + 2)}})\)

\(< 2 ^ {18} + 1 + (2 ^ {3 ^ {(2^{18} + 2)}} \uparrow \uparrow \uparrow (2^{18} + 4) ) \) (using Corollary 6)

\( < (2 \uparrow \uparrow 7) \uparrow \uparrow \uparrow (2^{18} + 4)\)

\(\le 2 \uparrow \uparrow 2 \uparrow \uparrow \ldots 2 \uparrow \uparrow (7 + 2^{18} + 4 - 1) \) (using Lemma 7)

\(\le  2 \uparrow \uparrow 2 \uparrow \uparrow \ldots 2 \uparrow \uparrow (2 \uparrow \uparrow \uparrow 4)\)

\(= 2 \uparrow \uparrow \uparrow (2^{18} + 8) \).

\( k_3 = k_2 + HJ(k_2 + 2, 2 ^ {(2+1) ^ {k_2 + 3 - 2 - 1}})  <  2 \uparrow \uparrow \uparrow (2^{18} + 8) + HJ ( (2 \uparrow \uparrow \uparrow (2^{18} + 8)) + 2, 2 ^ {3 ^ { 2 \uparrow \uparrow \uparrow (2^{18} + 8)}})\)

\(< 2 \uparrow \uparrow \uparrow (2^{18} + 8) + ( 2 ^ {3 ^ { 2 \uparrow \uparrow \uparrow (2^{18} + 8)}} \uparrow \uparrow \uparrow ((2 \uparrow \uparrow \uparrow (2^{18} + 8)) + 2)\)

\(< (2 \uparrow \uparrow \uparrow (2^{18} + 9) )  \uparrow \uparrow \uparrow ((2 \uparrow \uparrow \uparrow (2^{18} + 8)) + 2)\)

\(= (2 \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 8) ) \uparrow \uparrow \uparrow ((2 \uparrow \uparrow \uparrow (2^{18} + 8)) + 2)\)

\(\le 2 \uparrow \uparrow 2 \uparrow \uparrow 2 \ldots 2 \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 8) + 2 \uparrow \uparrow \uparrow (2^{18} + 8) + 1) \) with \(2 \uparrow \uparrow \uparrow (2^{18} + 8) + 2 \) 2's (using Lemma 7)

\( < 2 \uparrow \uparrow 2 \uparrow \uparrow 2 \ldots 2 \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 9) )\)

\(= 2 \uparrow \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 8) + 2 + 2^{18} + 9)\)

\(= 2 \uparrow \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 9)). \)

So \( GR(2, 2, 1, 2) \le k_3 < 2 \uparrow \uparrow \uparrow (2 \uparrow \uparrow \uparrow (2^{18} + 9)) = 2 \uparrow \uparrow \uparrow (2 \uparrow \uparrow \uparrow 262153) \) is an upper bound to the Graham's number problem.  Note that this number is comparable to \(G_1\), and far less than even \(G_2\).

General bounds for G(n, t, l, c):

A general bound for \( RAM(t, l, c) \) is \(c \uparrow c \uparrow \ldots c \uparrow (2ct) \) with \(l \) terms in the tower. Thus we apply the  Hales-Jewett Theorem  \(c \uparrow c \uparrow \ldots c \uparrow (2ct) \) times, and obtain an upper bound of about:

\( n \uparrow \uparrow \uparrow \uparrow (c \uparrow c \uparrow \ldots c \uparrow (2ct)) \) with \(l \) terms in the exponential tower. Deedlit11 (talk) 01:32, March 7, 2013 (UTC)