Hypergraph Tur\'{a}n densities can have arbitrarily large algebraic degree

TL;DR

This paper demonstrates that for any integer r ≥ 3 and any degree d, there exists a finite family of r-graphs with Turán density having algebraic degree at least d, answering a longstanding open question.

Contribution

It proves that Turán densities of r-graphs can have arbitrarily large algebraic degrees, extending Grosu's question to a broad class of densities.

Findings

Existence of finite r-graph families with Turán density of arbitrary algebraic degree.

Answer to Grosu's question in a strong form.

Turán densities can have unbounded algebraic complexity.

Abstract

Grosu [On the algebraic and topological structure of the set of Tur\'{a}n densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Tur\'{a}n density, as a real number, has (algebraic) degree greater than~$r-1$. In this note we show that, for all integers $r\ge 3$ and $d$, there exists a finite family of $r$-graphs whose Tur\'{a}n density has degree at least~$d$, thus answering Grosu's question in a strong form.