Mythical tree problem

The mythical tree problem is a combinatorial problem created by Harvey Friedman.
In the problem, the tree has a strange growth pattern. In the first stage, the tree grows k branches, forming k treetops. In the next stage, one of the treetops forms k+1 branches, increasing the number of treetops. The problem asks: what is the maximum number of branch segments a tree starting at k branches grow at once, during the tree's final stage of growth, provided that a squirrel can go from the root to any treetop without navigating more than n branch segments?
Friedman considers the ternary function V(k,m,n) defined as follows:
V(1,n,m) = n
V(k+1,n,1) = V(k,n+1,n+1) 
V(k+1,n,m+1) =
V(k,V(k+1,n,m)+1,V(k+1,n,m)+1).
Define seg(n,k) as the maximum number of branch segments as above. Friedman has shown that seg(n,k) = V(n,k,k).
From that, he has shown that:
* n = 4 and k = 2 corresponds to seg(4,2) = 22,539,988,369,406 segments (there was a misprint in his work: there are exactly 41*239 - 2 branch segments, not 40*239 - 2)
* n = 4 and k = 3 corresponds to a value seg(4,3) > 2295
* n = 5 and k = 2 corresponds to a value seg(5,2) > 2 ↑↑ 2295
* The growth rate of the number of segments as a function of k can be generalized to be similar to that of the Ackermann function.
== External links ==
* Friedman's posting at FOM
=== See also ===
Category:Functions
Category:Theorems
Category:Up-arrow notation level