On set-like sunflower-free families of subspaces over finite fields
Abstract
The Erdős--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 Blokhuis, De Boeck and D'haeseleer. In this note, we focus on the set-like setting. {We first observe by illustrations that general-position sunflower-freeness does not imply set-like sunflower-freeness.} We then present a construction of set-like s-sunflower-free families of k-spaces \revised{in 𝔽qn for ⌈(n+1)/2⌉ ≤ k ≤ n−1, of size qn−k (and qn−k+1 for even q),} based on a manipulated version of the lifting construction; to our knowledge, this is the first systematic construction tailored to this setting. Finally, we compare the size of the construction with the exact optimum for small parameters, computed by exhaustive search.
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BibTeX
@misc{otal2026setlike,
title = {{On set-like sunflower-free families of subspaces over finite fields}},
author = {Kamil Otal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.12232},
eprint = {2605.12232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12232},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On set-like sunflower-free families of subspaces over finite fields
AU - Kamil Otal
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2605.12232
UR - https://arxiv.org/abs/2605.12232
AB - The Erdős--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 Blokhuis, De Boeck and D'haeseleer. In this note, we focus on the set-like setting. {We first observe by illustrations that general-position sunflower-freeness does not imply set-like sunflower-freeness.} We then present a construction of set-like s-sunflower-free families of k-spaces \revised{in 𝔽_q^n for ⌈(n+1)/2⌉ ≤ k ≤ n−1, of size q^n−k (and q^n−k+1 for even q),} based on a manipulated version of the lifting construction; to our knowledge, this is the first systematic construction tailored to this setting. Finally, we compare the size of the construction with the exact optimum for small parameters, computed by exhaustive search.
ER -