On the Existence of a \((2, 3)\)-Spread in \(V(7, 2)\)

Abstract

An \((s, t)\)-spread in a finite vector space \(V = V(n, q)\) is a collection \(\mathcal{F}\) of \(t\)-dimensional subspaces of \(V\) with the property that every \(s\)-dimensional subspace of \(V\) is contained in exactly one member of \(F\). It is remarkable that no \((s, t)\)-spreads have been found yet, except in the case \(s = 1\).
In this note, the concept of an \(\alpha\)-point to a \((2,3)\)-spread \(\mathcal{F}\) in \(V = V(7,2)\) is introduced. A classical result of Thomas, applied to the vector space \(V\), states that all points of \(V\) cannot be \(\alpha\)-points to a given \((2,3)\)-spread \(\mathcal{F}\) in \(V\). In this note, we strengthen this result by proving that every \(6\)-dimensional subspace of \(V\) must contain at least one point that is not an \(\alpha\)-point to a given \((2, 3)\)-spread.