On $t$-intersecting Families of Spanning Trees

TL;DR

This paper establishes an optimal upper bound on the size of collections of spanning trees in a complete graph where every pair shares at least t edges, extending previous results to larger t values.

Contribution

It proves a tight bound for the maximum size of t-intersecting spanning tree families in complete graphs, improving earlier bounds for larger t.

Findings

Bound is tight and achieved by fixing t disjoint edges.

Extends previous results to larger t values.

Provides a new combinatorial bound for intersecting spanning trees.

Abstract

We prove that there exists a constant $c>0$ such that for all integers $2\leq t\leq cn$, if $\calA$ is a collection of spanning trees in $K_n$ such that any two intersect at at least $t$ edges, then $|\calA|\leq 2^tn^{n-t-2}$. This bound is tight; the equality is achieved when $\calA$ is a collection of spanning trees containing a fixed $t$ disjoint edges. This is an improvement of a result by Frankl, Hurlbert, Ihringer, Kupavskii, Lindzey, Meagher, and Pantagi, who proved such a result for $t=O\left(\frac n{\log n}\right)$.