The Erdős-Rado Sunflower Problem for Vector Spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914043633074176 |
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| author | Ihringer, Ferdinand Kupavskii, Andrey |
| author_facet | Ihringer, Ferdinand Kupavskii, Andrey |
| contents | 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 $s \geq k+1$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Erdős-Rado Sunflower Problem for Vector Spaces Ihringer, Ferdinand Kupavskii, Andrey Combinatorics 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 $s \geq k+1$, 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. |
| title | The Erdős-Rado Sunflower Problem for Vector Spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.03671 |