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arXiv · 2010.00726

Hypergraph regularity and higher arity VC-dimension

Artem Chernikov·Henry Towsner
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Abstract

We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels---one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples---and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1. We also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices.

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Artem Chernikov, Henry Towsner. 2020-10-01. Hypergraph regularity and higher arity VC-dimension. https://arxiv.org/abs/2010.00726

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