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Define SCGD(n) as the maximal length of sequence of graphs (G_{1}, G_{2}, ..., G_{m}) such that
 The degree of every vertex of every G_{i} is at most 3.
 For all i, G_{i} has at most n+i vertices.
 For all i < j, G_{i} is not graph minor of G_{j}.
 For all i, the diamond graph (K_{4} with one edge removal) is not graph minor of G_{i}.
SCGD function grows much slower than SCG function, but not slower than TREE function.
Here I'll show encoding of ncolored trees (the objects of TREE function) into subcubic graphs without diamond graph as minor (the objects of SCGD function).
The root (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)
A nonroot node (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)
To decoâ€¦
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Catching hierarchy fails beyond \(C_1(\Omega)\) (or \(C_{\Omega_2}(\Omega)\)).
Recall the definition:
 \(C_\pi(0)=\psi_\pi(\Omega_\omega)\).
 If \(C_\pi(\alpha)=\psi_\pi(\beta)\), then \(C_\pi(\alpha+1)=\psi_\pi(\gamma)\) where \(\psi(\gamma)\) is the least ordinal that \(g_{\psi(\gamma)}\) is comparable to \(f_{\psi(\gamma)}\) and \(\gamma>\beta\), and both \(\psi_\pi(\beta)\) and \(\psi_\pi(\gamma)\) are fullsimplified.
 \(\pi\) is the diagonalizer of \(C_\pi()\) function.
 For limit \(\alpha\), \(C_\pi(\alpha)[n]=C_\pi(\alpha[n])\).
But what's \(C_1(\Omega)\)? The \(\Omega\) works as the diagonalizer of C(), which is outside, so \(C_1(\Omega)\) refers to the outside C().
Now consider \(C_1(\Omega+1)\). \(C_1(\Omega+1)=\psi_1(\gamma)\) where \(\psâ€¦
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I don't know how you understand "a monotonic FS system", but in this blog post, I'll use "monotonic" for some special FS systems.
Definition 1: In a fundamental sequence system \(S:\mu\cap\text{Lim}\times\mathbb N\rightarrow\mu\), let \(\lambda\alpha\ge0\).
 If \(\beta\) is a successor ordinal, then let \(\beta=\gamma+1=\lambda[n_1]_S[n_2]_S\cdots[n_k]_S[]_S\) where \(n_k>0\) (\(k\ge0\)). \(\alpha

Here are two extensions of subcubic graph function, but I don't know whether they are welldefined.
Graph A is called a graph minor of graph B if A can be obtained from B by these 3 operations:
 Delete an edge;
 Delete an isolated vertex;
 Edge contraction  for an edge ab, delete this edge, and replace vertex a and b by one vertex connecting to the edges which a or b formerly connect to. (If a and b are linked by another edge, it will become a selfloop after edge contraction.)
Graph A is called a topological minor of graph B if A can be obtained from B by these 3 operations:
 Delete an edge;
 Delete an isolated vertex;
 Smoothing  for a (degree=2) vertex a which links to b and c, delete vertex a, and replace edge ab and ac by one edge bc.
For (unlabeled) subcubic graphs, graph mâ€¦
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In definitions of some ordinal collapsing functions (OCFs), there is always a series of \(C_n(\text{some arguments})\) sets. \(C_0(\text{something})\) may contains some argument in the "something", and may contains a "large" ordinal for collapsing; \(C_{n+1}(\text{something})\) is usually obtained from \(C_n(\text{something})\) by applying some operations with some limitations (avoiding "loop" definition); and \(C(\text{something})\) is the union of all \(C_n(\text{something})\). By this structure, we can define general fundamental sequences for OCFs.
Here this "general" definition works on the notation using a weakly compact cardinal. The definition of the ordinal notation is:
 Let \(K\) denote the weakly compact cardinal, \(\Omega_0=0\) andâ€¦
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