Lagrangians are attained as uniform Tur\'an densities

TL;DR

This paper proves that the set of uniform Turán densities for 3-graphs is not well-ordered, and shows that every Lagrangian of a 3-graph scaled by t/6 is attainable as a uniform Turán density.

Contribution

It establishes that the set of uniform Turán densities is not well-ordered and connects Lagrangians of 3-graphs to attainable densities.

Findings

The set of uniform Turán densities is not well-ordered.

Every scaled Lagrangian t/6 of a 3-graph is a uniform Turán density.

Disproves Erdős's jumping conjecture in the context of uniform Turán densities.

Abstract

The study of uniform Tur\'an densities was initiated in the 1980s by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an 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 $(d+o(1))\binom{\vert S\vert}{3}$ edges must contain a copy of $F$. Disproving Erd\H{o}s's famous jumping conjecture, Frankl and R\"odl showed that the set of Tur\'an densities is not well-ordered. We prove an analogous result for the uniform Tur\'an density, namely that the set $\Pi^{(3)}_{\therefore,\infty}=\{\pi_{\therefore}(\mathcal{F}) : \mathcal{F}\text{ a family of }3\text{-graphs} \}$ 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 $1 \leq t \leq 6$ we have $\frac{t}{6}\Lambda\in \Pi^{(3)}_{\therefore,\infty}$.