A single $3$-graph with infinite stability number

TL;DR

This paper constructs a simple explicit 3-graph with an infinite stability number, demonstrating that no finite set of structures can approximate all near-extremal configurations avoiding a single forbidden family.

Contribution

It extends the infinite-stability phenomenon from finite forbidden families to a single 3-graph, advancing understanding of extremal combinatorics.

Findings

Constructed a 3-graph with infinite stability number

Extended infinite-stability results to single-forbidden setting

Showed coexistence of many extremal constructions with stability

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.