We prove an Erdős–Ko–Rado-type theorem for
intersecting k-chains of subspaces of a finite
vector space. This is the q-generalization of earlier results of
Erdős, Seress and Székely for intersecting k-chains of subsets
of an underlying set. The proof hinges on the author's
proper generalization of the shift technique from extremal set theory to finite vector spaces,
which uses a linear map to define the generalized shift operation. The theorem is the
following.
For c = 0, 1, consider k-chains of subspaces of an
n-dimensional vector space over GF(q), such that the smallest
subspace in any chain has dimension at least c, and the
largest subspace in any chain has dimension at most n − c. The
largest number of such k-chains under the condition that any two share
at least one subspace as an element of the chain, is achieved by the
following constructions:
(1) fix a subspace of dimension c and take all k-chains containing it,
(2) fix a subspace of dimension n − c and take all
k-chains containing it.