On possible uniform Turán densities

TL;DR

This work develops a comprehensive palette-based framework to study uniform Turán densities for 3-graphs. It proves that every palette Lagrangian Λ_P is realized as the uniform density of some finite family, thereby embedding Λ_{pal} into Π_{x,fin} and implying the existence of irrational uniform Turán densities for finite families. The authors establish a palette regularity theory with a palette removal lemma, a Ramsey-type distinguishing result, and a stability argument showing that extremal palettes must be blow-ups of P or rev(P). Together, these components culminate in a main theorem linking palette Lagrangians to finite-family uniform densities and aligning with Lamaison’s approximation results, thereby clarifying the structure of finite-family uniform Turán densities and their relationship to Lagrangians.

Abstract

Given a family of $3$-graphs $\mathcal{F}$, the uniform Turán density $π_{\therefore}(\mathcal{F})$ 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 $Π_{\therefore,\text{fin}}$ 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 $Π_{\therefore,\text{fin}}$ contains every $d$ that is obtained as the uniform density of an optimized palette construction. A corollary of this is that $Π_{\therefore,\text{fin}}$ 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 $Π_{\therefore,\text{fin}}$ can be approximated by uniform densities of palette constructions.

Figures (3)

• Figure 4.1: An example with $n=2$. The ordered graph $A^0_1$ is in red and $A^0_2$ is in blue. On the second line we have the Ramsey graph $A^1_1$ with several copies of $A^0_1$ all intersecting in either an edge or a single vertex. Finally, in the last line we have the graph $A^1_2$ on the same set of vertices.
• Figure 4.2: An example of $G_{\sigma}$ for the permutation $\sigma\in S_4$ given by $\sigma(1)=3$, $\sigma(2)=1$, $\sigma(3)=4$ and $\sigma(4)=2$ and $a_\sigma=2$, $b_{\sigma}=3$, $c_\sigma=1$ and $d_\sigma=2$. The edges are given by $e_1=\{1,2,4,6\}$ (green), $e_2=\{3,4,5,7\}$ (blue) and $e_3=\{2,5,8,9\}$ (red).
• Figure 4.3: An example of $G$ for the palette $Q=\{q_1,q_2,q_3,q_4\}$ given by $q_1=(\text{blue}, \text{green}, \text{blue})$, $q_2=(\text{blue},\text{red},\text{red})$, $q_3=(\text{green}, \text{green}, \text{blue})$ and $q_4=(\text{red},\text{blue},\text{green})$. The graph $G$ consists of $m=4$ triangles and it can be partitioned into $G_{\text{blue}}\cup G_{\text{green}}\cup G_{\text{red}}$ as shown in the picture.

Theorems & Definitions (77)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• proof
• Proposition 2.5
• proof
• Theorem 2.6
• Lemma 3.1
• Definition 3.2
• Lemma 3.3: Palette Removal Lemma