On $t$-intersecting Families of Spanning Trees

We prove that there exists a constant $c>0$ such that for all integers $2\le t\le cn$, if $\text{calA}$ is a collection of spanning trees in $K_n$ such that any two intersect at at least $t$ edges, then $\mathrm{\mid }\text{calA}\mathrm{\mid }\le 2^t{n}^{n-t-2}$. This bound is tight; the equality is achieved when $\text{calA}$ 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 $t=O\left(\frac{n}{\log n}\right)$.