A Polynomial Ramsey Statement for Bounded VC-dimension

TL;DR

This paper proves a polynomial bound for a Ramsey-type property in bipartite graphs with bounded VC-dimension, extending previous results and implications for finite model theory.

Contribution

It establishes a polynomial dependency for a Ramsey statement in graphs of bounded VC-dimension, leveraging recent Erdős-Hajnal property results.

Findings

Polynomial bound for bipartite graphs with bounded VC-dimension

Extension of Ding et al.'s theorem with polynomial dependency

Implications for finite model theory and restricted structures

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.