Work

Induced subgraph density. VI. Bounded VC-dimension

Year: 2025

Type: article

Abstract: We confirm a conjecture of Fox, Pach, and Suk, that for every d > 0 , there exists c > 0 such that every n -vertex graph of VC-dimension at most d has a clique or stable set of size at least n c . This implies that, in the language of model theory, every graph definable in NIP structures has a clique or anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko, and Thomas. Our result also implies that every two-colourable tournament satisfies the tournament version of the Erdős-Hajnal conjecture, which completes the verification of the conjecture for six-vertex tournaments. The result extends to uniform hypergraphs of bounded VC-dimension as well. The proof method uses the ultra-strong regularity lemma for graphs of bounded VC-dimension proved by Lovász and Szegedy and the method of iterative sparsification introduced by the authors in an earlier paper.

Source: Advances in Mathematics

DOI: 10.1016/j.aim.2025.110601

Authors: Tung Nguyen, Alex Scott, Paul Seymour

Institutions: Princeton University, University of Oxford

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,790

Funders: National Science Foundation, National Sleep Foundation, Engineering and Physical Sciences Research Council, Air Force Office of Scientific Research