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Intersecting Chains in Finite Vector Spaces

    • Published online: 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 nc. 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 nc and take all k-chains containing it.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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