An irrational Lagrangian density of a single hypergraph

Abstract

The {\em Tur\'an 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 Tur\'{a}n density} of $F$ is defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ For graphs, Erd\H{o}s-Stone-Simonovits (\cite{ESi}, \cite{ES}) showed that $\Pi_{\infty}^{(2)}=\Pi_{fin}^{(2)}=\Pi_{1}^{(2)}=\{0, {1 \over 2}, {2 \over 3}, \ldots,{l-1 \over l}, ...\}.$ We know quite few about the Tur\'an 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 ${\sqrt 3\over 3}$, consequently, the Tur\'an density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.