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

TL;DR

This paper improves the sunflower bound for sets of pairwise intersecting $k$-spaces in projective spaces over large finite fields, showing such sets are sunflowers under a new, tighter size condition.

Contribution

The authors derive a sharper bound for when a set of pairwise intersecting $k$-spaces in PG(n,q) must be a sunflower, specifically for q ≥ 9, refining previous classical bounds.

Findings

The new bound applies for q ≥ 9.

Sets larger than the bound are necessarily sunflowers.

The bound depends on q and the dimension parameters.

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 $q\geq 9$: a set $\mathcal{S}$ of $k$-spaces in PG(n,q), $q\geq 9$, pairwise intersecting in a point is a sunflower if $|\mathcal{S}|> \left(\frac{2}{\sqrt[6]{q}}+\frac{4}{\sqrt[3]{q}}-\frac{5}{\sqrt{q}}\right)\left(\frac {q^{k + 1} - 1}{q - 1}\right)^2$.