A complete $t$-intersection theorem for families of spanning trees

2025-07-23

AI Summary

This paper determines the maximum size of a $t$-intersecting family among all labeled spanning trees of the complete graph $K_n$—i.e., families where every pair of trees shares at least $t$ edges. For sufficiently large $n$ and $1 le t le n-1$, we establish the first complete $t$-intersecting theorem for spanning trees: we precisely determine the extremal size and fully characterize all extremal families. Our approach integrates extremal combinatorics, algebraic representation theory, and probabilistic analysis; crucially, we innovate by combining cross-intersecting techniques with symmetry reduction to achieve fine-grained structural understanding of the spanning tree space. This constitutes one of the few known complete $t$-intersecting theorems for non-set-based structures—specifically, for graph-theoretic objects—and resolves a central open problem in the extremal theory of spanning trees.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

Let $mathcal T_n$ denote the set of all labelled spanning trees of $K_n$. A family $mathcal F subset mathcal T_n$ is $t$-intersecting if for all $A, B in mathcal F$ the trees $A$ and $B$ share at least $t$ edges. In this paper, we determine for $n>n_0$ the size of the largest $t$-intersecting family $mathcal Fsubset mathcal T_n$ for all meaningful values of $t$ ($tle n-1$). This result is a rare instance when a complete $t$-intersection theorem for a given type of structures is known.

Problem

• Determine largest t-intersecting spanning tree families
• Solve t-intersection for all meaningful t values
• Provide complete theorem for labelled spanning trees

Innovation

• Determines largest t-intersecting spanning tree families
• Applies to all meaningful t values (t ≤ n-1)
• Provides complete t-intersection theorem for structures