On the sunflower bound for k-spaces, pairwise intersecting in a point

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space $$\mathrm {PG}(n,q)$$ , where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a code is a sunflower if $$|C| > \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) ^2 + \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1$$ . In this article we will look at the case $$t=0$$ and we will improve this bound for $$q\ge 9$$ : a set $$\mathcal {S}$$ of k-spaces in $$\mathrm {PG}(n,q), q\ge 9$$ , pairwise intersecting in a point is a sunflower if $$|\mathcal {S}|> \left( \frac{2}{\root 6 \of {q}}+\frac{4}{\root 3 \of {q}}- \frac{5}{\sqrt{q}}\right) \left( \frac{q^{k + 1} - 1}{q - 1}\right) ^2$$ .