Hypergraphs with irrational Turán density and many extremal configurations

Abstract

Unlike graphs, determining Turán densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many near-extremal ${\mathcal{M}}_t$-free configurations with very different structure. Such a phenomenon is called not stable, and Liu and Mubayi gave a first not stable example. Another perhaps reason is that little is known about the set consisting of all possible Turán densities which has cardinality of the continuum. Let $t\ge 2$ be an integer. In this paper, we construct a finite family ${\mathcal{M}}_t$ of 3-uniform hypergraphs such that the Turán density of ${\mathcal{M}}_t$ is irrational, and there are $t$ near-extremal ${\mathcal{M}}_t$-free configurations that are far from each other in edit-distance. This is the first not stable example that has an irrational Turán density. It also provides a new phenomenon about feasible region functions.

arXiv admin note: text overlap with arXiv:2206.03948