For t < c n the only maximal (t-1)-intersection-free families in GL(n,q) are the t-umvirates and their duals.
The forbidden cross intersection problem for permutations.arXiv preprint arXiv:2512.11372,
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Forbidden Intersection Theorems for Matrix Spaces
For t < c n the only maximal (t-1)-intersection-free families in GL(n,q) are the t-umvirates and their duals.