A hypergraph Tur\'{a}n problem with no stability

TL;DR

This paper constructs a hypergraph family with a known Turán number, demonstrating that near-extremal configurations can differ greatly, thus showing the absence of a stability theorem in this context.

Contribution

It introduces a finite family of triple systems with a determined Turán number and proves the existence of far-apart near-extremal constructions, breaking the stability paradigm.

Findings

Two near-extremal constructions are far in edit-distance.

The family $\\mathcal{M}$ has a precisely determined Turán number.

First example of a hypergraph family lacking a stability theorem.

Abstract

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Tur\'an's conjecture. Our main result is to construct a finite family of triple systems $\mathcal{M}$, determine its Tur\'{a}n number, and prove that there are two near-extremal $\mathcal{M}$-free constructions that are far from each other in edit-distance. This is the first extremal result for a hypergraph family that fails to have a corresponding stability theorem.