An irrational Lagrangian density of a single hypergraph

Abstract

The {\em 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 {\em Turán density} of $F$ is defined as $\pi (F)=\underset{n\to \infty }{\lim }\frac{ex(n,F)}{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}.$ For graphs, Erdős-Stone-Simonovits (\cite{ESi}, \cite{ES}) showed that ${\Pi }_{\infty }^{(2)}={\Pi }_{fin}^{(2)}={\Pi }_1^{(2)}=\{0,\frac{1}{2},\frac{2}{3},\dots ,\frac{l-1}{l},...\}.$ We know quite few about the Turán density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot \cite{BT}, and Pikhurko \cite{Pikhurko2} showed that there is an irrational number in ${\Pi }_3^{(3)}$ and ${\Pi }_{fin}^{(3)}$ respectively, disproving a conjecture of Chung and Graham \cite{FG}. Baber and Talbot \cite{BT} asked whether ${\Pi }_1^{(r)}$ contains an irrational number. In this paper, we show that the Lagrangian density of $F=\{123,124,134,234,567\}$ (the disjoint union of $K_4^3$ and an edge) is $\frac{\sqrt{3}}{3}$, consequently, the Turán density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.