my forty years on his shoulders - Department of Mathematics

42THEOREM 8.10. (finite KT). Let n >> k. For all finite trees T 1 ,...,T n with each |T i |≤ i+k, there exists i < j such that T i is inf preserving embeddable into T j .Since Theorem 8.10 → Theorem 8.9 → Corollary 8.5 (using bounds n+k,variable n, k constant), we see that Theorem 8.10 is not provable in FS.Other Π 0 2 forms of KT involving only the internal structure of a singlefinite tree can be found in (Friedman 2002b).We proved analogous results for EKT = extended Kruskal theorem, whichinvolves a finite label set and a gap embedding condition. Only here the strengthjumps up to that of Π 1 1-CA 0 .We said that the gap condition was natural (i.e., EKT was natural). Manypeople were unconvinced.Soon later, EKT became a tool in the proof of the well known graph minortheorem of Robertson, Seymour (Robertson, Seymour 1985, 2004).THEOREM 8.11. Let G 1 ,G 2 ,... be finite graphs. There exists i < j such that G i isminor included in G j .We then asked Robertson and Seymour to prove a form of EKT that weknew implied full EKT, just from GMT. They complied, and we wrote the triplepaper (Friedman, Robertson, Seymour 1987).The upshot is that GMT is not provable in Π 1 1-CA 0 . Just where GMT isprovable is unclear, and recent discussions with Robertson have not stabilized.We disavow remarks in (Friedman, Robertson, Seymour 1987) about where GMTcan be proved.