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 edges in $F$-free on $n$ vertices. density $F$ defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ Denote $\Pi_{\infty}^{(r)}={ \pi(\cal F): \cal F a family r{-uniform graphs}},$ $\Pi_{fin}^{(r)}=\{ {is \ finite of} r{{-}uniform graphs}\}$, and $\Pi_{t}^{(r)}=\{\pi(\cal graphs, and}|\cal F|\le t}.$ For Erdös Simonovits [Studia Sci. Mat. Hungar. 1 (1966), pp. 51--57] Stone [Bull. Amer. Math. Soc., 52 (1946), 1087--1091] showed that $\Pi_{\infty}^{(2)}=\Pi_{fin}^{(2)}=\Pi_{1}^{(2)}={0, {1 2}, {2 3}, ...,{l-1 l}, ...}.$ We know quite little about for $r\ge 3$. Baber Talbot [Electron. J. Combin., 19 (2011)] Pikhurko [Israel Math., 20 (2014), 415--454] there irrational $\Pi_{3}^{(3)}$ $\Pi_{fin}^{(3)}$, respectively, disproving conjecture Chung Graham [Erdös Graphs: His Legacy Unsolved Problems, A. K. Peters, Natick, MA, 1999]. asked whether $\Pi_{1}^{(r)}$ contains number. Lagrangian hypergraph has been useful tool extremal problems. $\pi_{\lambda}(F)=\sup \{r! \lambda(G):G\;is;F-free\}$, where $\lambda(G)$ $G$. Sidorenko [Combinatorica, 9 (1989), 207--215] same extension $F$. In this paper, we show $F={123, 124, 134, 234, 567}$ (the disjoint union $K_4^3$ edge) ${\sqrt 3\over 3}$, consequently, number, answering question Talbot.