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

Abstract

A $t$-intersecting constant dimension subspace code $C$ is a set of $k$-dimensional subspaces in a projective space PG(n,q), where distinct subspaces intersect in 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 PG(n,q), $q\ge 9$, pairwise intersecting in a point is a sunflower if $|\mathcal{S}|>\left(\frac{2}{\sqrt[6]{q}}+\frac{4}{\sqrt[3]{q}}-\frac{5}{\sqrt{q}}\right){\left(\frac{q^{k+1}-1}{q-1}\right)}^2$.