Unlike graphs, determining Tur\'{ a}n 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 Tur\'{ a}n 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 Tur\'{ a}n 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 Tur\'{ a}n density. It also provides a new phenomenon about feasible region functions.
Categories: math.CO