On the existence of a \((2,3)\)-spread in \(V(7,2)\). (Q2828900)

Let \(\mathbb{F}_q\) be the finite field with \(q\) elements. An \((s,t)\)-spread in \(\mathbb{F}_q^n\) is a collection \(\mathcal{F}\) of \(t\)-dimensional subspaces of \(\mathbb{F}_q^n\) such that every \(s\)-dimensional subspace of \(\mathbb{F}_q^n\) is contained in exactly one element of \(\mathcal{F}\). In general, \((1,t)\)-spreads are simply called \(t\)-spreads. It has been conjectured that for \(s>1\) no \((s,t)\)-spread exists.NEWLINENEWLINEIn this paper, the authors deal with the smallest open case, that is \((2,3)\)-spread in \(\mathbb{F}_{2}^7\) and they give a necessary condition for the existence of such objects. To this aim, they call a point \(P\) an \(\alpha\)-point to \(\mathcal{F}\) if every \(5\)-dimensional subspace \(T\) of \(\mathbb{F}_2^7\) which contains two elements of \(\mathcal{F}\) meeting at \(P\) is such that all its five members from \(\mathcal{F}\) contain \(P\).NEWLINENEWLINE The main result is the following.NEWLINENEWLINE Theorem. If \(\mathcal{F}\) is a \((2,3)\)-spread in \(\mathbb{F}_2^7\) then every \(6\)-dimensional subspace of \(\mathbb{F}_2^7\) contains at least one point which is not an \(\alpha\)-point to \(\mathcal{F}\).