A Polynomial Ramsey Statement for Bounded VC-dimension
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917099077632000 |
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| author | Hons, Tomáš |
| author_facet | Hons, Tomáš |
| contents | A theorem by Ding, Oporowski, Oxley, and Vertigan states that every sufficiently large bipartite graph without twins contains a matching, co-matching, or half-graph of any given size as an induced subgraph. We prove that this Ramsey statement has polynomial dependency assuming bounded VC-dimension of the initial graph, using the recent verification of the Erdős-Hajnal property for graphs of bounded VC-dimension. Since the theorem of Ding et al. plays a role in (finite) model theory, which studies even more restricted structures, we also comment on further refinements of the theorem within this context. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Polynomial Ramsey Statement for Bounded VC-dimension Hons, Tomáš Combinatorics 05D10 G.2.2 A theorem by Ding, Oporowski, Oxley, and Vertigan states that every sufficiently large bipartite graph without twins contains a matching, co-matching, or half-graph of any given size as an induced subgraph. We prove that this Ramsey statement has polynomial dependency assuming bounded VC-dimension of the initial graph, using the recent verification of the Erdős-Hajnal property for graphs of bounded VC-dimension. Since the theorem of Ding et al. plays a role in (finite) model theory, which studies even more restricted structures, we also comment on further refinements of the theorem within this context. |
| title | A Polynomial Ramsey Statement for Bounded VC-dimension |
| topic | Combinatorics 05D10 G.2.2 |
| url | https://arxiv.org/abs/2502.20461 |