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Published online by Cambridge University Press: 01 November 1999
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.