A survey of Zarankiewicz problems in geometry
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910878425677824 |
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| author | Smorodinsky, Shakhar |
| author_facet | Smorodinsky, Shakhar |
| contents | One of the central topics in extremal graph theory is the study of the function $ex(n,H)$, which represents the maximum number of edges a graph with $n$ vertices can have while avoiding a fixed graph $H$ as a subgraph. Tur{á}n provided a complete characterization for the case when $H$ is a complete graph on $r$ vertices. Erd{\H o}s, Stone, and Simonovits extended Tur{á}n's result to arbitrary graphs $H$ with $χ(H) > 2$ (chromatic number greater than 2). However, determining the asymptotics of $ex(n, H)$ for bipartite graphs $H$ remains a widely open problem. A classical example of this is Zarankiewicz's problem, which asks for the asymptotics of $ex(n, K_{t,t})$.
In this paper, we survey Zarankiewicz's problem, with a focus on graphs that arise from geometry. Incidence geometry, in particular, can be viewed as a manifestation of Zarankiewicz's problem in geometrically defined graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03702 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A survey of Zarankiewicz problems in geometry Smorodinsky, Shakhar History and Overview Computational Geometry Discrete Mathematics Combinatorics One of the central topics in extremal graph theory is the study of the function $ex(n,H)$, which represents the maximum number of edges a graph with $n$ vertices can have while avoiding a fixed graph $H$ as a subgraph. Tur{á}n provided a complete characterization for the case when $H$ is a complete graph on $r$ vertices. Erd{\H o}s, Stone, and Simonovits extended Tur{á}n's result to arbitrary graphs $H$ with $χ(H) > 2$ (chromatic number greater than 2). However, determining the asymptotics of $ex(n, H)$ for bipartite graphs $H$ remains a widely open problem. A classical example of this is Zarankiewicz's problem, which asks for the asymptotics of $ex(n, K_{t,t})$. In this paper, we survey Zarankiewicz's problem, with a focus on graphs that arise from geometry. Incidence geometry, in particular, can be viewed as a manifestation of Zarankiewicz's problem in geometrically defined graphs. |
| title | A survey of Zarankiewicz problems in geometry |
| topic | History and Overview Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2410.03702 |