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 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 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, much larger than TREE(3). 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)."^[3]

See also

• Goodstein's theorem
• Paris–Harrington theorem
• Kanamori–McAloon theorem
• Kruskal's tree theorem
• Robertson–Seymour theorem

References