Intersecting Families of Spanning Trees of $K_{n,n}$

A family of spanning trees of a graph is $t$-intersecting if any pair of spanning trees in the family has $t$ or more edges in common. For sufficiently large $n$ and $t\le n\mathrm{/}C{\log }_{2}n$ for some absolute constant $C>0$, we give a nearly complete characterization of the extremal $t$-intersecting families of spanning trees in balanced complete bipartite graphs with parts of order $n$. In particular, for $t=1$, we give exact bounds and a full characterization of the extremal families. For $t\ge 2$, our bounds are tight up to lower-order terms, and we show that any extremal $t$-intersecting family is of the form $\mathcal{F}\cup {\mathcal{S}}^{\mathrm{\prime }}$ where $\mathcal{F}$ is a family of all trees containing a fixed $t$-matching, and ${\mathcal{S}}^{\mathrm{\prime }}$ is a distinguished set of exceptional trees of size $\mathrm{\mid }{\mathcal{S}}^{\mathrm{\prime }}\mathrm{\mid }=o(\mathrm{\mid }\mathcal{F}\mathrm{\mid })$.