On possible uniform Turán densities

Abstract

Given a family of $3$-graphs $\mathcal{F}$, the uniform Turán density ${\pi }_{\therefore }(\mathcal{F})$ is defined as the infimum $d\in [0,1]$ for which any sufficiently large uniformly $d$-dense $3$-graph - that is, a $3$-graph which has edge-density at least $d$ on all linearly sized subsets - contains a copy of some $F\in \mathcal{F}$. Let ${\Pi }_{\therefore ,\text{fin}}$ denote the set of all possible uniform Turán densities of finite families. Erdős, Hajnal, and Rödl introduced a family of constructions for lower bounds on uniform Turán densities called palette constructions. We show that ${\Pi }_{\therefore ,\text{fin}}$ contains every $d$ that is obtained as the uniform density of an optimized palette construction. A corollary of this is that ${\Pi }_{\therefore ,\text{fin}}$ contains the set of Lagrangians of $3$-graphs and includes irrational numbers. Our work complements a recent result of Lamaison, which states that every value in ${\Pi }_{\therefore ,\text{fin}}$ can be approximated by uniform densities of palette constructions.