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