For intersecting families of rank-k flats in the graphic matroid of K_{n+1}, size is at most binom(n-1,k-1) whenever n+1 >= 8k, with equality only for a full edge-star.
Kupavskii, Erd˝ os–Ko–Rado type results for partitions via spread approximations,European J
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Erd\H{o}s--Ko--Rado and Hilton--Milner Theorems in the Partition Lattice
For intersecting families of rank-k flats in the graphic matroid of K_{n+1}, size is at most binom(n-1,k-1) whenever n+1 >= 8k, with equality only for a full edge-star.