A dichotomy for hypergraph Zarankiewicz problems on axis-parallel boxes
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908987435253760 |
|---|---|
| author | Chao, Ting-Wei Dong, Zichao Liu, Hong Shu, Xichao Wang, Shuaichao |
| author_facet | Chao, Ting-Wei Dong, Zichao Liu, Hong Shu, Xichao Wang, Shuaichao |
| contents | We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends the problems studied by Chan and Har-Peled on points and $d$-dimensional boxes in $\mathbb{R}^d$, corresponding to $(F_1,F_2)=(\varnothing,[d])$, as well as by Chan, Keller, and Smorodinsky on $r$ families of $d$-dimensional boxes, corresponding to $(F_1,\dots,F_r)=([d],\dots,[d])$.
Our main result establishes a sharp dichotomy for the Zarankiewicz number in this setting: it is either $Θ_r(tn^{r-1})$ or at least $Ω\bigl( tn^{r-1} \cdot \frac{\log n}{\log\log n} \bigr)$, depending only on a simple set-theoretic condition on $(F_1,\dots,F_r)$, which we call $2$-coherence. Informally, $2$-coherence captures whether the configuration contains an underlying two-dimensional incidence structure, which is precisely what gives rise to the extra polylogarithmic factor. Our proof proceeds via a sequence of reductions and a geometric slicing argument that reduces the problem to planar incidence bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A dichotomy for hypergraph Zarankiewicz problems on axis-parallel boxes Chao, Ting-Wei Dong, Zichao Liu, Hong Shu, Xichao Wang, Shuaichao Combinatorics We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends the problems studied by Chan and Har-Peled on points and $d$-dimensional boxes in $\mathbb{R}^d$, corresponding to $(F_1,F_2)=(\varnothing,[d])$, as well as by Chan, Keller, and Smorodinsky on $r$ families of $d$-dimensional boxes, corresponding to $(F_1,\dots,F_r)=([d],\dots,[d])$. Our main result establishes a sharp dichotomy for the Zarankiewicz number in this setting: it is either $Θ_r(tn^{r-1})$ or at least $Ω\bigl( tn^{r-1} \cdot \frac{\log n}{\log\log n} \bigr)$, depending only on a simple set-theoretic condition on $(F_1,\dots,F_r)$, which we call $2$-coherence. Informally, $2$-coherence captures whether the configuration contains an underlying two-dimensional incidence structure, which is precisely what gives rise to the extra polylogarithmic factor. Our proof proceeds via a sequence of reductions and a geometric slicing argument that reduces the problem to planar incidence bounds. |
| title | A dichotomy for hypergraph Zarankiewicz problems on axis-parallel boxes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.20815 |