For n large and 2≤t≤n−2, every t-intersecting family of spanning trees of K_n has size at most c_{n,t} n^{n−2−t}, with equality exactly for the trivial family containing a balanced fixed forest.
Filmus,The weighted complete intersection theorem, Journal of Combinatorial Theory, Series A 151 (2017), 84–101
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A complete $t$-intersection theorem for families of spanning trees
For n large and 2≤t≤n−2, every t-intersecting family of spanning trees of K_n has size at most c_{n,t} n^{n−2−t}, with equality exactly for the trivial family containing a balanced fixed forest.