A single $3$-graph with infinite stability number
Combinatorics
2026-05-22 v1
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 -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--graph direction of Balogh--Clemen--Luo, in which exponentially many exact extremal constructions coexist with stability.
Keywords
Cite
@article{arxiv.2605.21877,
title = {A single $3$-graph with infinite stability number},
author = {Heng Li and Xizhi Liu},
journal= {arXiv preprint arXiv:2605.21877},
year = {2026}
}