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Keyword: uniform hypergraphs ×Clear all

2 claims from 1 paper

1. Mathematics › Limits and Structures in Graph Theory

   Induced subgraph density. VI. Bounded VC-dimension

   Nguyen, Scott and Seymour · arXiv (Cornell University) · 2023

   The paper proves that graphs of bounded VC-dimension have polynomial-size cliques or stable sets, with consequences in model theory and for tournaments, using a regularity lemma and iterative sparsification.

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   1. UncheckedGraphs of bounded VC-dimension on n vertices always contain a clique or stable set of size at least n to the power c, for some c depending on the dimension.“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 $n^c$.”
   2. Unchecked“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.”