An irrational Lagrangian density of a single hypergraph

The Turán number of an r-uniform graph F , denoted by ex(n, F ), is the maximum number of edges in an F -free r-uniform graph on n vertices. The Turán density of F is defined as π(F ) = lim n→∞ ex(n,F ) (nr) . Denote Π (r) ∞ = {π(F) : F is a family of r−uniform graphs}, Π fin = {π(F) : F is a finite family of r−uniform graphs} and Π t = {π(F) : F is a family of r−uniform graphs and |F| ≤ t}. For graphs, Erdős-Stone-Simonovits ([7], [8]) showed that Π ∞ = Π fin = Π (2) 1 = {0, 1 2 , 2 3 , ..., l−1 l , ...}. We know quite few about the Turán density of an r-uniform graph for r ≥ 3. Baber and Talbot [2], and Pikhurko [27] showed that there is an irrational number in Π (3) 3 and Π (3) fin respectively, disproving a conjecture of Chung and Graham [5]. Baber and Talbot [2] asked whether Π (r) 1 contains an irrational number. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an r-uniform graph F is πλ(F ) = sup{r!λ(G) : G is F -free}, where λ(G) is the Lagrangian of an r-uniform graph G. Sidorenko [31] showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turán density of the extension of F . In this paper, we show that the Lagrangian density of F = {123, 124, 134, 234, 567} (the disjoint union of K 4 and an edge) is √ 3 3 , consequently, the Turán density of the extension of F is an irrational number, answering the question of Baber and Talbot.