A single $3$-graph with infinite stability number

Heng Li; Xizhi Liu

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arXiv · 2605.21877

A single $3$-graph with infinite stability number

Heng Li·Xizhi Liu

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Abstract

The stability number of a forbidden family measures how many different structures are needed to approximate all near-extremal constructions avoiding it. An infinite stability number means that no finite list of structures suffices. We construct a simple explicit $3$-graph whose stability number is infinite. This extends the infinite-stability phenomenon for finite forbidden families, established by Hou--Li--Liu--Mubayi--Zhang, to the single-forbidden setting, and further develops the single-$3$-graph direction of Balogh--Clemen--Luo, in which exponentially many exact extremal constructions coexist with stability.

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Heng Li, Xizhi Liu. 2026-05-21. A single $3$-graph with infinite stability number. https://arxiv.org/abs/2605.21877

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