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arXiv · 2312.15572

Induced subgraph density. VI. Bounded VC-dimension

Tung Nguyen·Alex Scott·Paul Seymour
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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 Erd\H{o}s-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 Lov\'asz and Szegedy and the method of iterative sparsification introduced by the authors in an earlier paper.

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Tung Nguyen, Alex Scott, Paul Seymour. 2023-12-25. Induced subgraph density. VI. Bounded VC-dimension. https://arxiv.org/abs/2312.15572

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