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) denotes that length for simple subcubic graphs. The function SCG(k) 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 − 8 ≈ 3.241704 ⋅ 1035775080127201286522908640066 and its decimal expansion ends in ...11352349133049430008. SSCG(3) is much, much larger than TREE(3). Adam P. 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)."
- Goodstein's theorem
- Paris–Harrington theorem
- Kanamori–McAloon theorem
- Kruskal's tree theorem
- Robertson–Seymour theorem