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Induced subgraph density. VI. Bounded VC-dimension
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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.
Forward citations
Cited by 2 Pith papers
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An Alon-Boppana--type bound for very dense graphs, with applications to max-cut
A new spectral argument shows that very dense graphs far from unions of cliques have least eigenvalue at most -n^{1/4-epsilon}, and this yields max-cut surpluses of order n^{1.01} and m^{0.5001} for H-free graphs.
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Interpolating chromatic and homomorphism thresholds
The authors determine the exact VC-dimension-interpolated homomorphism thresholds for cliques and prove the blowup threshold of odd cycles C_{2k-1} is 1/(2k-1).
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