2025/04/29 by King, Dylan, Piga, Simón, Sales, Marcelo +1 · 1 citation
#05C65 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.21220
Given a family of 3-graphs F, the uniform Turán density π\therefore(F) is defined as the infimum d∈[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 ∈ F. Let Π\therefore,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,fin contains every d that is obtained as the uniform density of an optimized palette construction. A corollary of this is that Π\therefore,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,fin can be approximated by uniform densities of palette constructions.