For large n and moderate t, the largest t-intersecting family of spanning trees of K_n is the family of all trees containing a fixed set of t disjoint edges, with stars added when t=1.
The Erd˝ os–Ko–Rado theorem for vector spaces
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Intersecting Families of Spanning Trees
For large n and moderate t, the largest t-intersecting family of spanning trees of K_n is the family of all trees containing a fixed set of t disjoint edges, with stars added when t=1.