Friedman's SSCG function

In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has a degree of 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. Then, by applying Kőnig's lemma on the tree of such sequences under extension, 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 slower than SCG, SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 2^{(3 × 295)} − 8 ≈ 3.241704 × 10^{35775080127201286522908640065}. Its first and last 20 digits are 32417042291246009846...34057047399148290040. SSCG(3) is much larger than both TREE(3) and TREE^TREE(3)(3), that is, the TREE function nested TREE(3) times with 3 at the bottom.

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]

The function was proposed and studied by Harvey Friedman.

See also

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

References

1. [FOM] 274:Subcubic Graph Numbers
2. [FOM] 279:Subcubic Graph Numbers/restated
3. TREE(3) and impartial games | Complex Projective 4-Space