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arXiv · 1707.02643

The Junta Method for Hypergraphs and the Erd\H{o}s-Chv\'{a}tal Simplex Conjecture

Nathan Keller·Noam Lifshitz
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Abstract

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a $k$-uniform hypergraph on $n$ vertices that does not contain an `enlarged' copy $H^+$ of a fixed hypergraph $H$. These include well-known problems such as the Erd\H{o}s-S\'{o}s `forbidding one intersection' problem and the Frankl-F\"{u}redi `special simplex' problem. We present a general approach to such problems, using a `junta approximation method' that originates from analysis of Boolean functions. We prove that any $H^+$-free hypergraph is essentially contained in a `junta' -- a hypergraph determined by a small number of vertices -- that is also $H^+$-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all $C n_0(d)$.

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Nathan Keller, Noam Lifshitz. 2017-07-09. The Junta Method for Hypergraphs and the Erd\H{o}s-Chv\'{a}tal Simplex Conjecture. https://arxiv.org/abs/1707.02643

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