On set-like sunflower-free families of subspaces over finite fields

Abstract

The Erd\H{o}s--Rado sunflower problem admits two natural analogues in finite vector spaces, corresponding to two different ways of generalising the set-theoretic notion of a sunflower. The first, used by Ihringer and Kupavskii [FFA 110 (2026) 102746], requires the petals to be in general position over the kernel; the second, used in the subspace codes literature (cf.\ Etzion--Raviv [DAM 186 (2015) 87-97], Blokhuis--De Boeck--D'haeseleer [DCC 90 (2022) 2101-2111]), requires only that the kernel equals the pairwise intersection of distinct petals. We refer to the second version as a \emph{set-like sunflower}, following Ihringer and Kupavskii. In this note, we focus on the set-like setting. We observe that the constructions of Ihringer--Kupavskii, although correct under their (stronger) definition, do not yield set-like sunflower-free families: we exhibit explicit set-like sunflowers inside their Example~3.1. We then present a construction of set-like $s$-sunflower-free families of $k$-spaces, based on a manipulated version of the lifting construction. To our knowledge, this is the first systematic construction tailored to this setting.