Intersection theorems for vector spaces
Let \(b\) be a positive integer, \({\mathcal F}=\{F_1,\ldots, F_m\}^a \) collection of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space over a finite field \(K\). Let there exist numbers \(\mu_1,\ldots, \mu_s\) with \(k\not\equiv \mu_i \pmod b\), \(1\leq t\leq s\), and for all \(i,j\) with \(1\leq i<j\leq m\) and some \(t\), \(\dim \vert F_i\cap F_j\vert \equiv \mu_t \pmod b\). Then \(\vert\mathcal F\vert \leq \left[ \begin{matrix} n \\ s\end{matrix} \right]_q\), with \[ \left[ \begin{matrix} n\\ s\end{matrix} \right]_ q = \prod_{0\leq i<s} \frac{q^{n-i}-1} {q^{t-i}-1}\quad\text{and}\quad \vert K\vert =q. \] Applying this a constructive lower bound is given for the Ramsey number \(R(k,k)\).
scientific article; zbMATH DE number 3922867