An Irrational Turán Density via Hypergraph Lagrangian Densities

Baber and Talbot asked whether there is an irrational Turán density of a single hypergraph. In this paper, we show that the Lagrangian density of a 4-uniform matching of size 3 is an irrational number. Sidorenko showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turán density of the extension of F. Therefore, our result gives a positive answer to the question of Baber and Talbot. We also determine the Lagrangian densities of a class of r-uniform hypergraphs on n vertices with θ(n2) edges. As far as we know, for every hypergraph F with known hypergraph Lagrangian density, the number of edges in F is less than the number of its vertices.