Changes: Subcubic graph number

Revision as of 03:15, 11 August 2018

The subcubic graph numbers are the outputs of a fast-growing combinatorial function.^[1] They were devised by Harvey Friedman, who showed that it eventually dominates every recursive function provably total in the theory of \(\Pi^1_1\)-\(\text{CA}_0\), and is itself provably total in the theory of \(\Pi_1^1-\text{CA}+\text{BI}\).

One output of the sequence, SCG(13), is a subject of extensive research. It is known to surpass TREE(3), a number that arises from a related sequence.

Definition

A subcubic graph is a finite graph in which each vertex has a valence of at most three. (For the sake of this article, subcubic graphs are allowed to be multigraphs, and are not required to be connected.) We also define the graph minor relation as follows: A is a graph minor of B if we can derive A from the following process: start with B, remove vertices and edges, and contract edges.^[2] An example of a graph minor derivation is shown in the infobox of this article.

Given an integer k, suppose we have a sequence of subcubic graphs G₁, G₂, ... such that each graph G_i has at most i + k vertices and for no i < j is G_i homeomorphically embeddable into G_j (i.e. is a graph minor).

The Robertson-Seymour theorem proves that subcubic graphs are well-quasi-ordered by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of k, there is a sequence with maximal length. We denote this maximal length using SCG(k).

Specific values

It is possible to show that SCG(0) = 6. The first graph is one vertex with a loop,

the second is two vertices connected by a single edge, and the next four graphs consist of 3, 2, 1, and 0 unconnected vertices. All maximal sequences will peak and decline this way.

The following bounds have been claimed by Googology Wiki user Hyp cos.

\(\text{SCG}(1) > f_{\varepsilon_22}(f_{\varepsilon_02}(f_{\varepsilon_0+1}(f_{\varepsilon_0}(f_{\omega^\omega+1}(f_{\omega^5+\omega^2+\omega}(\\f_{\omega^23+1}(f_{\omega^22+1}(f_{\omega^2+\omega3+1}(f_{\omega^2+1}(f_{\omega^2}(3\times2^{3\times2^{95}})))))))))))\).

\(\text{SCG}(2) > f_{\vartheta(\Omega^\omega)}(f_{\varepsilon_22}(f_{\varepsilon_02}(f_{\varepsilon_0+1}(f_{\varepsilon_0}(f_{\omega^\omega+1}(\\f_{\omega^5+\omega^2+\omega}(f_{\omega^23+1}(f_{\omega^22+1}(f_{\omega^2+\omega3+1}(f_{\omega^2+1}(f_{\omega^2}(3\times2^{3\times2^{95}}))))))))))))\)

These bounds use a non-standard choice of fundamental sequences for ordinals — by using a particular, highly complex bijection between ordinals and small graphs, which we will denote here by \(f\), we define \(\alpha[n]=\max\{\beta: \beta<\alpha\text{ and } f(\beta)\text{ is a graph with }\leq n\text{ vertices}\}\).

Since the graph indices start at one, it is also valid to say that SCG(-1) = 1, consisting only of the empty graph.

Friedman proved that SCG(13) is greater than the halting time of any Turing machine such that it can be proven to halt in at most ^2,0002 symbols in \(\Pi^1_1\)-\(\text{CA}_0\). It is therefore far larger than TREE(3).

SCG(n) is computable,therefore it is naturaly surpassed by \(\Sigma(n)\) for some n.

Matrix interpretation

An alternate way of describing the SCG function is as follows. Define an incidence matrix as a matrix with entries in {0, 1, 2} where each column sums to exactly 2 and each row sums to at most 3. Given a nonnegative integer k, we construct a sequence of n incidence matrices M₁, M₂, ..., M_n such that each matrix M_i has at most i + k rows, and no matrix can be changed into an earlier one by repeated applications of any of the following processes:

Reordering rows or columns.
Deleting columns.
Deleting rows, then deleting all columns that do not sum to 2.
Take two rows A and B such that A + B contains a 2 in position i for some i. Remove A and B, append A + B to the matrix, and finally remove column i.

SCG(k), then, is the largest possible value of n.

Simple subcubic graph numbers

If we require the subcubic graphs to be simple (i.e. no loops or multiple edges), we get the simple subcubic graph numbers, denoted SSCG. Adam P. Goucher has shown that SSCG(2) << TREE(3) << SSCG(3).^[3] Moreover, he has shown that even TREEⁿ(3) for even very large n (for example n=TREE(3)) does not compete at all with SSCG(3).

SCG(n) and SSCG(n) have comparable growth rates: Goucher proved that \(SSCG(4n+3) \geq SCG(n)\).^[4]

Values and bounds

SSCG(0) = 2
SSCG(1) = 5
SSCG(2) \(\geq 3 \cdot 2^{3 \cdot 2^{95}}-8 \approx 10^{3.5775 \cdot 10^{28}}\)

Sources

↑ Harvey Friedman, FOM 279:Subcubic Graph Numbers/restated
↑ Technically a topological minor, but topological minors and graph minors are equivalent for subcubic graphs.
↑ Adam Goucher, Graph minors
↑ Adam Goucher, TREE(3) and impartial games (see comment)

@@ Line 12: / Line 12: @@
 == Definition ==
-A '''subcubic graph''' is a finite graph in which each vertex has a valence of at most three. (For the sake of this article, subcubic graphs are allowed to be {{w|multigraph}}s, and are not required to be connected.) We also define the '''graph minor''' relation as follows: ''A'' is a graph minor of ''B'' iff we can derive ''A'' from the following process: start with ''B'', remove vertices and edges, and contract edges.<ref>Technically a topological minor, but topological minors and graph minors are equivalent for subcubic graphs.</ref> An example of a graph minor derivation is shown in the infobox of this article.
+A '''subcubic graph''' is a finite graph in which each vertex has a valence of at most three. (For the sake of this article, subcubic graphs are allowed to be {{w|multigraph}}s, and are not required to be connected.) We also define the '''graph minor''' relation as follows: ''A'' is a graph minor of ''B'' if we can derive ''A'' from the following process: start with ''B'', remove vertices and edges, and contract edges.<ref>Technically a topological minor, but topological minors and graph minors are equivalent for subcubic graphs.</ref> An example of a graph minor derivation is shown in the infobox of this article.
 Given an integer ''k'', suppose we have a sequence of subcubic graphs G<sub>1</sub>, G<sub>2</sub>, ... such that each graph G<sub>i</sub> has at most ''i'' + ''k'' vertices and for no ''i'' < ''j'' is G<sub>''i''</sub> {{w|graph minor|homeomorphically embeddable}} into G<sub>''j''</sub> (i.e. is a graph minor).