Exact thresholds for graph minors

This note is part of implementation of a program in foundations of mathematics to find exact versions of all unprovability theorems known so far, a program initiated by A. Weiermann. In this note we find the exact version of unprovability of the graph minor theorem restricted to planar graphs and some lower and upper bounds in the general case of all graphs. 1 Unlabelled growth constants Let us fix a class of simple unlabelled graphs G. The classes we have in mind are planar graphs and connected planar graphs. We shall denote the number of n-vertex members of G as gn. We say that the class G has an unlabelled growth constant γG if (gn) →n→∞ γG. If G is the class of all planar graphs then γG exists [3] and is a number between 27.2269 and 30.061. It has been proved in [8] that for any given surface, the set of all unlabelled graphs embeddable into this surface also has an unlabelled growth constant and this constant coincides with the unlabelled planar growth constant. It has been conjectured that every class of unlabelled graphs omitting a given set of minors has an unlabelled growth constant, as was proved in the labelled case in [2]. 2 Approximation Lemma Let us first prove an approximation lemma that we shall need for the threshold result below. Lemma 1. Let gn be the number of n-vertex unlabelled planar graphs, γ be the unlabelled planar growth constant and Ck be the circle on k vertices. Denote the number of n-vertex planar graphs omitting the minor Ck by gn,k. Then for every k ∈ ω, there is γk such that 1. for every k ∈ ω, gn,k ≥ γ k ; 2. γ = supk∈ω γk. Proof. Let us show, by a superadditivity argument, that for every k, (gn,k) →n→∞ sup n (gn,k).