Friedman's SSCG function

In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has degree at most three. Suppose we have a sequence of simple subcubic graphs G₁, G₂, ... such that each graph G_i has at most i + k vertices (for some integer k) and for no i < j is G_i homeomorphically embeddable into (i.e. is a graph minor of) G_j.

The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of k, there is a sequence with maximal length. The function SSCG(k)^[1] denotes that length for simple subcubic graphs. The function SCG(k)^[2] denotes that length for (general) subcubic graphs.

The SCG sequence begins SCG(0) = 6, but then explodes to a value equivalent to f_ε2*2 in the fast-growing hierarchy.

The SSCG sequence begins SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 2^{(3 × 295)} − 8 ≈ 3.241704 × 10^{35775080127201286522908640066} and its decimal expansion ends in ...11352349133049430008.

SSCG(3) is much larger than both TREE(3) and TREE^TREE(3)(3).^[a] Adam P. Goucher claims there is no qualitative difference between the asymptotic growth rates of SSCG and SCG. He writes "It's clear that SCG(n) ≥ SSCG(n), but I can also prove SSCG(4n + 3) ≥ SCG(n)."^[3]

See also[]

• Goodstein's theorem
• Paris–Harrington theorem
• Kanamori–McAloon theorem

Notes[]

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