Lagrangians are attained as uniform Turán densities

Abstract

The study of uniform Turán densities was initiated in the 1980s by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density of $F$, ${\pi }_{\therefore }(F)$, is defined as the infimum $d\in [0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\left(\genfrac{}{}{0ex}{}{|S|}{3}\right)$ edges must contain a copy of $F$. Disproving Erdős's famous jumping conjecture, Frankl and Rödl showed that the set of Turán densities is not well-ordered. We prove an analogous result for the uniform Turán density, namely that the set ${\Pi }_{\therefore ,\infty }^{(3)}=\{{\pi }_{\therefore }(\mathcal{F}):\mathcal{F}\text{ a family of }3\text{-graphs}\}$ is not well-ordered. This is a consequence of a more general result, which in particular implies that for every Lagrangian $\Lambda$ of a $3$-graph and integer $1\le t\le 6$ we have $\frac{t}{6}\Lambda \in {\Pi }_{\therefore ,\infty }^{(3)}$.