The Erdős-Ko-Rado theorem for vector spaces
The classical Erdős-Ko-Rado problem was studied in many aspects: In 1975 W. N. Hsieh proved the first analogue theorem in finite vector spaces. In this case two subspaces are intersecting, if their common part in properly subspace. Let V be an n-dimensional vector space over GF(q) and for integers \(k\geq t>0\) let \(m_ q(n,k,t)\) denote the maximum possible number of subspaces in a t-intersecting family \({\mathcal F}\) of k- dimensional subspaces of V i.e. dim \(F\cap F'\geq t\) holds for all F,F'\(\in {\mathcal F}\). Hshieh proved in 1975 that \(m_ q(n,k,t)=\left[ \begin{matrix} n-t\\ k-t\end{matrix} \right]\) holds for \(n\geq 2k+1\), \(q\geq 3\) and for \(n\geq 2k+2\), \(q=2.\) In the paper under review it is proved: Theorem 1: Suppose \(n\geq 2k-t\); \({\mathcal F}\subset \left[ \begin{matrix} V\\ k\end{matrix} \right]\) is t- intersecting then \[ | {\mathcal F}| \leq \max \{\left[ \begin{matrix} n-t\\ k-t\end{matrix} \right]_ q;\quad \left[ \begin{matrix} 2k-t\\ k\end{matrix} \right]_ q\}. \] The proof method is based on the ideas of the second author.
scientific article; zbMATH DE number 3985272