Friedman's SSCG function

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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.

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.

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