A Lower Bound for Kruskal's Weak Tree Function tree(3)

TL;DR

This paper establishes a new explicit lower bound for Kruskal's weak tree function at n=3, demonstrating rapid growth even for small arguments through combinatorial analysis.

Contribution

It provides the first explicit lower bound for tree(3), advancing understanding of the weak tree function's growth behavior.

Findings

Lower bound for tree(3) is at least 844,424,930,131,960

Constructed explicit sequence of trees satisfying weak tree function constraints

Shows weak tree function grows rapidly even at small arguments

Abstract

We establish an explicit lower bound for Kruskal's weak tree function at n=3, proving that tree(3) >= 844,424,930,131,960 = 3 * 2^48 - 8. This is achieved by constructing an explicit sequence of unlabeled rooted trees satisfying the constraints of the weak tree function and carefully analyzing the combinatorics of the "leg elimination" process. Our bound significantly exceeds previous estimates and demonstrates that even for small arguments, the weak tree function exhibits rapid growth.