Intersecting Chains in Finite Vector Spaces

• Published in: Combinatorics, probability & computing probability & computing, 1999-11, Vol.8 (6), p.509-528
• Main Author: CZABARKA, ÉVA
• Format: Article
• Language: English
• ISSN: 0963-5483; 1469-2163
• DOI: 10.1017/S0963548399004010

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.