The Erdős-Rado Sunflower Problem for Vector Spaces

The famous Erdős-Rado sunflower conjecture suggests that an $s$-sun-flower-free family of $k$-element sets has size at most $(Cs{)}^k$ for some absolute constant $C$. In this note, we investigate the analog problem for $k$-spaces over the field with $q$ elements. For $s\ge k+1$, we show that the largest $s$-sunflower-free family $\mathcal{F}$ satisfies $$1\le \mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{/}{q}^{(s-1)\left(\genfrac{}{}{0px}{}{k+1}{2}\right)-k}\le (q\mathrm{/}(q-1){)}^k\mathrm{.}$$ For $s\le k$, we show that $$q^{-\left(\genfrac{}{}{0px}{}{k+1}{2}\right)}\le \mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{/}{q}^{(s-1)\left(\genfrac{}{}{0px}{}{k+1}{2}\right)-k}\le (q\mathrm{/}(q-1){)}^k\mathrm{.}$$ Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.