# Degenerate Tur\'an densities of sparse hypergraphs

## TL;DR

This paper determines the asymptotic maximum edge densities of certain sparse hypergraphs, extending classical results and introducing new combinatorial and algebraic constructions for bounds.

## Contribution

It extends the understanding of degenerate Turán densities for hypergraphs, providing explicit formulas and bounds for new parameter ranges, and introduces novel combinatorial and algebraic methods.

## Key findings

- Exact value of (r,2,3) as 1/(r^2 - r) for r  4.
- Bounds for (r,k,3) when r > k  3, including explicit inequalities.
- New combinatorial and algebraic constructions, including strongly 3-perfect hashing.

## Abstract

For fixed integers $r>k\ge 2,e\ge 3$, let $f_r(n,er-(e-1)k,e)$ be the maximum number of edges in an $r$-uniform hypergraph in which the union of any $e$ distinct edges contains at least $er-(e-1)k+1$ vertices. A classical result of Brown, Erd\H{o}s and S\'os in 1973 showed that $f_r(n,er-(e-1)k,e)=\Theta(n^k).$ The degenerate Tur\'an density is defined to be the limit (if it exists) $$\pi(r,k,e):=\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}.$$ Extending a recent result of Glock for the special case of $r=3,k=2,e=3$, we show that $$\pi(r,2,3):=\lim_{n\rightarrow\infty}\frac{f_r(n,3r-4,3)}{n^2}=\frac{1}{r^2-r-1}$$ for arbitrary fixed $r\ge 4$. For the more general cases $r>k\ge 3$, we show that $$\frac{1}{r^k-r}\le\liminf_{n\rightarrow\infty}\frac{f_r(n,3r-2k,3)}{n^k}\le\limsup_{n\rightarrow\infty}\frac{f_r(n,3r-2k,3)}{n^k}\le \frac{1}{k!\binom{r}{k}-\frac{k!}{2}}.$$ The main difficulties in proving these results are the constructions establishing the lower bounds. The first construction is recursive and purely combinatorial, and is based on a (carefully designed) approximate induced decomposition of the complete graph, whereas the second construction is algebraic, and is proved by a newly defined matrix property which we call {\it strongly 3-perfect hashing}.