The Erd\H{o}s-Rado Sunflower Problem for Vector Spaces

TL;DR

This paper explores the vector space analog of the Erdős-Rado sunflower conjecture, establishing bounds on sunflower-free families of subspaces over finite fields using advanced coding theory techniques.

Contribution

It extends the sunflower problem to vector spaces over finite fields and provides bounds using lifted MRD codes, a novel approach in this context.

Findings

Established bounds for sunflower-free families when s ≥ k+1.

Established bounds for sunflower-free families when s ≤ k.

Used iterative construction with lifted MRD codes for lower bounds.

Abstract

The famous Erd\H{o}s-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 $s \geq k+1$, we show that the largest $s$-sunflower-free family $\mathcal{F}$ satisfies \[ 1 \leq |\mathcal{F}| / q^{(s-1) \binom{k+1}{2} - k} \leq (q/(q-1))^k. \] For $s \leq k$, we show that \[ q^{-\binom{k+1}{2}} \leq |\mathcal{F}| / q^{(s-1) \binom{k+1}{2} - k} \leq (q/(q-1))^k. \] Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.