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arXiv · 2206.03948

Hypergraphs with infinitely many extremal constructions

Jianfeng Hou·Heng Li·Xizhi Liu·Dhruv Mubayi·Yixiao Zhang
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Abstract

We give the first exact and stability results for a hypergraph Tur\'{a}n problem with infinitely many extremal constructions that are far from each other in edit-distance. This includes an example of triple systems with Tur\'{a}n density $2/9$, thus answering some questions posed by the third and fourth authors and Reiher about the feasible region of hypergraphs. Our results also provide extremal constructions whose shadow density is a transcendental number. Our novel approach is to construct certain multilinear polynomials that attain their maximum (in the standard simplex) on a line segment and then to use these polynomials to define an operation on hypergraphs that gives extremal constructions.

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Jianfeng Hou, Heng Li, Xizhi Liu, Dhruv Mubayi, Yixiao Zhang. 2022-06-08. Hypergraphs with infinitely many extremal constructions. https://arxiv.org/abs/2206.03948

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