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

Mark Giroux

TL;DR

This work addresses the problem of establishing a nontrivial lower bound for Kruskal's weak tree function by focusing on tree(3). It introduces an explicit sequence of unlabeled rooted trees and a detailed leg-elimination analysis, culminating in the Leg Elimination Formula $L(x) = 6 \cdot 2^{x} - 2x - 6$ and the concrete bound $tree(3) \ge 844{,}424{,}930{,}131{,}960 = 3 \cdot 2^{48} - 8$. The main result demonstrates rapid growth of the weak tree function even for small arguments and discusses its implications for tree(4) and the fast-growing hierarchy. The paper situates the bound among known values, compares with prior estimates, and highlights open questions about tightness and broader consequences in combinatorial proof theory.

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.

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

TL;DR

This work addresses the problem of establishing a nontrivial lower bound for Kruskal's weak tree function by focusing on tree(3). It introduces an explicit sequence of unlabeled rooted trees and a detailed leg-elimination analysis, culminating in the Leg Elimination Formula and the concrete bound . The main result demonstrates rapid growth of the weak tree function even for small arguments and discusses its implications for tree(4) and the fast-growing hierarchy. The paper situates the bound among known values, compares with prior estimates, and highlights open questions about tightness and broader consequences in combinatorial proof theory.

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.
Paper Structure (14 sections, 4 theorems, 10 equations)

This paper contains 14 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tree Representation
  4. Known Values
  5. Main Result
  6. The Initial Sequence
  7. The Leg Elimination Process
  8. Final Calculation
  9. Visual Proof of Lemma \ref{['lem:leg']}
  10. A Note on the Initial Embedding
  11. Discussion
  12. Comparison with Previous Estimates
  13. Upper Bounds and Exactness
  14. Implications for tree(4) and Beyond

Key Result

Proposition 6

$\mathrm{tree}(1) = 2$, achieved by the sequence: A root with one child (2 vertices), followed by a single root node (1 vertex).

Theorems & Definitions (11)

  • Definition 1: Rooted Tree
  • Definition 2: Infimum / Least Common Ancestor
  • Definition 3: Inf-Embedding
  • Definition 4: Weak Tree Function
  • Definition 5: Visual Tree Representation
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Lemma 9: Leg Elimination Formula
  • proof : Proof that $T_1 \not\leq T_2$
  • ...and 1 more