Overview
Research details the construction of a simple, explicit 3-graph characterized by an infinite stability number. This finding extends previous observations of infinite stability within finite forbidden families to a single-forbidden context.
Research Context
The stability number of a forbidden family quantifies the diversity of structures required to approximate near-extremal constructions that avoid that family. An infinite stability number indicates that a finite set of structures is insufficient for this approximation. Prior work by Hou–Li–Liu–Mubayi–Zhang established the existence of infinite stability in the context of finite forbidden families. Separately, the work of Balogh–Clemen–Luo explored a single-3-graph direction, where multiple exact extremal constructions can coexist with stability. The current research builds upon these areas by addressing the single-forbidden setting within the 3-graph domain.
Approach
The methodology involved the construction of a specific 3-graph. The objective of this construction was to demonstrate that its stability number is infinite. The description of this 3-graph is given as "simple explicit."
Findings
The study reports the construction of a simple explicit 3-graph. This constructed 3-graph possesses an infinite stability number. This result signifies that, for this particular single-forbidden context, an approximation of near-extremal constructions would require an infinite list of diverse structures. The implication is that a finite set of structures is insufficient for such approximation in this specific setting.
Why This Matters
This finding extends the understanding of the infinite-stability phenomenon, moving its demonstration from finite forbidden families to the more constrained setting of a single forbidden structure. It further develops the study of single-3-graphs, particularly concerning the coexistence of exact extremal constructions and stability.