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arXiv · 2502.20461

A Polynomial Ramsey Statement for Bounded VC-dimension

Tomáš Hons
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Abstract

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 Erd\H{o}s-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.

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Tomáš Hons. 2025-02-27. A Polynomial Ramsey Statement for Bounded VC-dimension. https://arxiv.org/abs/2502.20461

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