Friedman’s SSCG function

In mathematics, a simple subcubic graph 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 Gi has at most i + k vertices (for some integer k) and for no i < j is Gi homeomorphically embeddable into (i.e. is a graph minor of) Gj.

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 SSCG sequence begins SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 23 × 295 − 9 ≈ 103.5775 × 1028. SSCG(3) is not only larger than TREE(3), it is much, much larger than TREE(TREE(…TREE(3)…))[3] where the total nesting depth of the formula is TREE(3) levels of the TREE function . Adam Goucher claims there’s 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)."[4]