Sunflowers and Ramsey problems for restricted intersections
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916699800862720 |
|---|---|
| author | Janzer, Barnabás Jin, Zhihan Sudakov, Benny Wu, Kewen |
| author_facet | Janzer, Barnabás Jin, Zhihan Sudakov, Benny Wu, Kewen |
| contents | Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set $L\subseteq \{0,\dots,k-1\}$ and a family $\mathcal{F}$ of $k$-element sets which does not contain a sunflower with $m$ petals whose kernel size is in $L$, how large a subfamily of $\mathcal{F}$ can we find in which no pair has intersection size in $L$? We give matching upper and lower bounds, determining the dependence on $m$ for all $k$ and $L$. This problem also finds applications in quantum computing.
As an application of our techniques, we also obtain a variant of Füredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sunflowers and Ramsey problems for restricted intersections Janzer, Barnabás Jin, Zhihan Sudakov, Benny Wu, Kewen Combinatorics Discrete Mathematics Quantum Physics Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set $L\subseteq \{0,\dots,k-1\}$ and a family $\mathcal{F}$ of $k$-element sets which does not contain a sunflower with $m$ petals whose kernel size is in $L$, how large a subfamily of $\mathcal{F}$ can we find in which no pair has intersection size in $L$? We give matching upper and lower bounds, determining the dependence on $m$ for all $k$ and $L$. This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of Füredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method. |
| title | Sunflowers and Ramsey problems for restricted intersections |
| topic | Combinatorics Discrete Mathematics Quantum Physics |
| url | https://arxiv.org/abs/2504.15264 |