Induced subgraph density. VI. Bounded VC-dimension
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918138415677440 |
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| author | Nguyen, Tung Scott, Alex Seymour, Paul |
| author_facet | Nguyen, Tung Scott, Alex Seymour, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15572 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Induced subgraph density. VI. Bounded VC-dimension Nguyen, Tung Scott, Alex Seymour, Paul Combinatorics 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. |
| title | Induced subgraph density. VI. Bounded VC-dimension |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.15572 |