Degenerate Turán densities of sparse hypergraphs

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, Erdős and Sós in 1973 showed that $f_r(n,er-(e-1)k,e)=\mathrm{\Theta }(n^k)$. The degenerate Turán density is defined to be the limit (if it exists)

$$\pi (r,k,e):=\underset{n\to \infty }{\lim }\frac{f_r(n,er-(e-1)k,e)}{n^k}.$$

Extending a recent result of Glock for the special case of $r=3,k=2,e=3$, we show that

$$\pi (r,2,3):=\underset{n\to \infty }{\lim }\frac{f_r(n,3r-4,3)}{n^2}=\frac{1}{r^2-r-1}$$

for arbitrary fixed $r\ge 4$. For the more general cases $r>k\ge 3$, we manage to show

$$\frac{1}{r^k-r}\le \underset{n\to \infty }{\lim \inf }\frac{f_r(n,3r-2k,3)}{n^k}\le \underset{n\to \infty }{\lim \sup }\frac{f_r(n,3r-2k,3)}{n^k}\le \frac{1}{k!\left(\begin{array}{c}r\\ k\end{array}\right)-\frac{k!}{2}},$$

where the gap between the upper and lower bounds are small for $r\gg k$.

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 strongly 3-perfect hashing.

Introduction

Turán-type problems have been playing a central role in the field of extremal graph theory since Turán [40] determined in 1941 the Turán number of complete graphs. In this work we focus on a classical hypergraph Turán-type problem introduced by Brown, Erdős and Sós [10] in 1973.

For an integer $r\ge 2$, an r-uniform hypergraph $\mathcal{H}$ (or r-graph, for short) on the vertex set $V(\mathcal{H})$, is a family of r-element subsets of $V(\mathcal{H})$, called the edges of $\mathcal{H}$. An r-graph is said to contain a copy of $\mathcal{H}$ if it contains $\mathcal{H}$ as a subhypergraph. Furthermore, given a family $\mathcal{H}$ of r-graphs, an r-graph is said to be $\mathcal{H}$-free if it contains no copy of any member of $\mathcal{H}$. The Turán number ${\mathrm{ex}}_r(n,\mathcal{H})$, is the maximum number of edges in an $\mathcal{H}$-free r-graph on n vertices. It can be easily shown that the sequence ${\{{\left(\left(\begin{array}{c}n\\ r\end{array}\right)\right)}^{-1}\cdot {\mathrm{ex}}_r(n,\mathcal{H})\}}_{n=r}^{\infty }$ is bounded and non-increasing, and therefore converges [28]. Hence, the Turán density $\pi (\mathcal{H})$ of $\mathcal{H}$ is defined to be

$$\pi (\mathcal{H}):=\underset{n\to \infty }{\lim }\frac{{\mathrm{ex}}_r(n,\mathcal{H})}{\left(\begin{array}{c}n\\ r\end{array}\right)}.$$

If $\pi (\mathcal{H})=0$ then $\mathcal{H}$ is called degenerate. It is well-known (see, e.g. [13], [29], [31]) that $\mathcal{H}$ is degenerate if and only if it contains an r-partite r-graph, where an r-graph is called r-partite if its vertex set admits a partition into r disjoint parts $V_1,\dots ,V_r$, such that every edge of it contains exactly one vertex from each vertex part $V_i$. If $\mathcal{H}$ is degenerate and there exists a real number $\alpha \in (0,r)$ such that ${\mathrm{ex}}_r(n,\mathcal{H})=\mathrm{\Theta }(n^{\alpha })$, then the degenerate Turán density ${\pi }_d(\mathcal{H})$ of $\mathcal{H}$ is defined to be the limit (if it exists)

$${\pi }_d(\mathcal{H}):=\underset{n\to \infty }{\lim }\frac{{\mathrm{ex}}_r(n,\mathcal{H})}{n^{\alpha }},$$

where α is called the Turán exponent¹ of $\mathcal{H}$. For example, it is known (see, e.g. [23]) that ${\pi }_d(C_4)={\lim }_{n\to \infty }\frac{{\mathrm{ex}}_2(n,C_4)}{n^{3/2}}=\frac{1}{2}$, where $C_4$ is the cycle of length 4.

For a positive integer n let $[n]:=\{1,\dots ,n\}$, and for any $X\subseteq [n]$ let $\left(\begin{array}{c}X\\ r\end{array}\right)$ be the family of $\left(\begin{array}{c}|X|\\ r\end{array}\right)$ distinct r-subsets of X. For fixed integers $r\ge 2,e\ge 2,v\ge r+1$, let ${\mathcal{G}}_r(v,e)$ be the family of all r-graphs formed by e edges and at most v vertices; that is,

$${\mathcal{G}}_r(v,e)=\{\mathcal{H}\subseteq \left(\begin{array}{c}[v]\\ r\end{array}\right):|\mathcal{H}|=e,|V(\mathcal{H})|\le v\}.$$

Thus an r-graph is ${\mathcal{G}}_r(v,e)$-free if and only if the union of any e distinct edges contains at least $v+1$ vertices. Since such r-graphs do not contain many edges (see (1) below), they are also termed sparse hypergraphs [22]. Following previous papers on this topic (see, e.g. [1]) we will use the notation

$$f_r(n,v,e):={\mathrm{ex}}_r(n,{\mathcal{G}}_r(v,e))$$

to denote the maximum number of edges in a ${\mathcal{G}}_r(v,e)$-free r-graph.

In 1973, Brown, Erdős and Sós [10] initiated the study of the function $f_r(n,v,e)$, which has attracted considerable attention throughout the years. More concretely, they showed that

$$\mathrm{\Omega }(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=O(n^{\lceil \frac{er-v}{e-1}\rceil }).$$

The lower bound was proved by a standard probabilistic argument (now known as the alteration method, see, e.g. Chapter 3 of [3]), and the upper bound follows from a double counting argument, which uses the simple fact that in a ${\mathcal{G}}_r(v,e)$-free r-graph, any set of $\lceil \frac{er-v}{e-1}\rceil$ vertices can be contained in at most $e-1$ distinct edges. Improvements on (1) for less general parameters were obtained in a series of works, see, e.g. [1], [9], [10], [12], [17], [14], [24], [25], [32], [33], [34], [35], [36], [38].

In this paper we are interested in the special case where $k:=\frac{er-v}{e-1}$ is an integer greater than one. In such a case the order of $f_r(n,v,e)$ is determined by (1), i.e.,

$$f_r(n,er-(e-1)k,e)=\mathrm{\Theta }(n^k),$$

where $v=er-(e-1)k$ and $2\le k\le r-1$. Thus for fixed integers $e\ge 2,r>k\ge 2$, it is natural to ask whether the limit

$${\pi }_d(r,k,e):={\pi }_d({\mathcal{G}}_r(er-(e-1)k,e))=\underset{n\to \infty }{\lim }\frac{f_r(n,er-(e-1)k,e)}{n^k}$$

exists, where we call ${\pi }_d(r,k,e)$ the degenerate Turán density of sparse hypergraphs.

For $e=2$ this question is already resolved, since an r-graph is ${\mathcal{G}}_r(2r-k,2)$-free if and only if any pair of its edges share at most $k-1$ vertices, therefore $f_r(n,2r-k,2)$ is equal to the maximum size of an $(n,r,k)$-packing, where an $(n,r,k)$-packing is a family of r-subsets of $[n]$ such that any k-subset of $[n]$ is contained in at most one member of this family. Clearly, the largest size of an $(n,r,k)$-packing cannot exceed $\left(\begin{array}{c}n\\ k\end{array}\right)/\left(\begin{array}{c}r\\ k\end{array}\right)$. Moreover, it was shown by Rödl [33] (see [26], [30] for the current state-of-the-art) that for fixed $r,k$ and sufficiently large n, this bound is essentially tight, up to a $1-o(1)$ factor (where $o(1)\to 0$ as $n\to \infty$). This implies that

$${\pi }_d(r,k,2)=\underset{n\to \infty }{\lim }\frac{(1-o(1))\left(\begin{array}{c}n\\ k\end{array}\right)/\left(\begin{array}{c}r\\ k\end{array}\right)}{n^k}=\frac{1}{r\cdots (r-k+1)}.$$

For $e\ge 3$ not much is known, and only recently the existence of ${\pi }_d(3,2,3)$ was resolved. Brown, Erdős and Sós [10] posed the following conjecture (see also [9]).

Conjecture 1 Brown, Erdős and Sós [10]

The degenerate Turán density

$${\pi }_d(3,2,e)=\underset{n\to \infty }{\lim }\frac{f_3(n,e+2,e)}{n^2}$$

exists for every fixed $e\ge 3$.

For the first case $e=3$, they were able to show that $1/6\le {\pi }_d(3,2,3)\le 2/9$. To the best of our knowledge, for more than forty years no significant improvement was made until recently Glock [25] closed the gap by showing that

$${\pi }_d(3,2,3)=\underset{n\to \infty }{\lim }\frac{f_3(n,5,3)}{n^2}=\frac{1}{5}.$$

In this paper we continue this line of research, and in the spirit of (2) and Conjecture 1 we consider the following question.

Question 2

For fixed integers $r>k\ge 2,e\ge 3$, does the limit

$${\pi }_d(r,k,e)=\underset{n\to \infty }{\lim }\frac{f_r(n,er-(e-1)k,e)}{n^k}$$

exist? If so, what is the value of ${\pi }_d(r,k,e)$?

In general this question is widely open. The authors of [10] who established (2) did not try to optimize the coefficient of $n^k$, however a careful analysis of their lower bound yields to

$$\sqrt[e-1]{\frac{(er-(e-1)k)!}{2\left(\begin{array}{c}er-(e-1)k\\ r\end{array}\right)\cdot \left(\begin{array}{c}\left(\begin{array}{c}er-(e-1)k\\ r\end{array}\right)\\ e\end{array}\right)\cdot {(r!)}^e}}\le {\pi }_d(r,k,e)\le \frac{e-1}{r\cdots (r-k+1)},$$

where the upper bound follows from the observation that any k-subset of $[n]$ is contained in at most $e-1$ edges of a ${\mathcal{G}}_r(er-(e-1)k,e)$-free r-graph, implying that $f_r(n,er-(e-1)k,e)\le (e-1)(\left(\begin{array}{c}n\\ k\end{array}\right)/\left(\begin{array}{c}r\\ k\end{array}\right))$. Note that (4) states in fact lower and upper bounds on ${\lim \inf }_{n\to \infty }\frac{f_r(n,er-(e-1)k,e)}{n^k}$ and ${\lim \sup }_{n\to \infty }\frac{f_r(n,er-(e-1)k,e)}{n^k}$ respectively, since it is not known whether ${\pi }_d(r,k,e)$ exists. However, to simplify the notations we keep (4) in its current form, and in the sequel we will frequently use abbreviations of this type.

Our main results are introduced in the next two subsections, and they include the determination of ${\pi }_d(r,2,3)$ for any fixed $r\ge 4$, and new lower and upper bounds for ${\pi }_d(r,k,3)$ for any fixed $r>k\ge 3$.

Notations. We use standard asymptotic notations $\mathrm{\Omega }(\cdot ),\mathrm{\Theta }(\cdot ),O(\cdot )$ and $o(\cdot )$ as $n\to \infty$, where for functions $f=f(n)$ and $g=g(n)$, we write $f=O(g)$ if there is a constant $c_1$ such that $|f|\le c_1|g|$; we write $f=\mathrm{\Omega }(g)$ if there is a constant $c_2$ such that $|f|\ge c_2|g|$; we write $f=\mathrm{\Theta }(g)$ if $f=O(g)$ and $f=\mathrm{\Omega }(g)$ hold simultaneously; finally, we write $f=o(g)$ if ${\lim }_{n\to \infty }(f/g)=0$.

In an $(n,r,2)$-packing, any member of $\left(\begin{array}{c}[n]\\ 2\end{array}\right)$ is contained in at most one member of $\left(\begin{array}{c}[n]\\ r\end{array}\right)$, therefore one can easily verify that such a packing is also ${\mathcal{G}}_r(3r-4,3)$-free. This implies that for all fixed $r\ge 4$ the result of Rödl [33], written in the above notation, is

$${\pi }_d(r,2,3)\ge \frac{1}{r^2-r}.$$

We will give a tighter bound than (5) by showing that approximately, a ($\frac{1}{r^2-r-1}$)-fraction of the 2-subsets in $[n]$ can be contained in two r-subsets, while the resulting hypergraph still has the ${\mathcal{G}}_r(3r-4,3)$-free property (see Remark 19). As a consequence, we obtain the following improvement on the above lower bound.

Theorem 3

For any fixed integer $r\ge 4$,

$$\pi (r,2,3)=\underset{n\to \infty }{\lim }\frac{f_r(n,3r-4,3)}{n^2}=\frac{1}{r^2-r-1}.$$

Note that Theorem 3 extends (3) from $r=3$ to arbitrary fixed $r\ge 4$. To prove this theorem it suffices to show that ${\lim \sup }_{n\to \infty }\frac{f_r(n,3r-4,3)}{n^2}\le \frac{1}{r^2-r-1}$ and ${\lim \inf }_{n\to \infty }\frac{f_r(n,3r-4,3)}{n^2}\ge \frac{1}{r^2-r-1}$. The upper bound is a special case of the upper bound stated in Theorem 6 below, which will be discussed later. The main difficulty in proving Theorem 3 is the construction which establishes the lower bound. In what follows we briefly review the main ideas behind it.

Generally speaking, the lower bound is obtained by a recursive construction (recursion on the uniformity r) and a carefully designed approximate induced decomposition of $K_n$, the complete graph on n vertices. Given a finite graph G, a G-packing in $K_n$ is simply a family of edge disjoint copies of G in $K_n$. We will make use of the following lemma, which was proved to be very useful in many other combinatorial constructions (see, e.g. [2], [4], [19], [21], [25]).

Lemma 4 Graph packing lemma, see Theorem 2.2 [19] or Theorem 3.2 [5]

Let G be any fixed graph with e edges and $ϵ>0$ be any small constant. Then there is an integer $n_0$ such that for any $n>n_0$, there exists a G-packing $\mathcal{G}=\{G^1,\dots ,G^l\}$ in $K_n$ with

$$l\ge (1-ϵ)\frac{n^2}{2e}$$

edge disjoint copies of G such that

• any two distinct copies of G share at most two vertices, i.e., $|V(G^i)\cap V(G^j)|\le 2$ for any $1\le i\ne j\le l$;

• if two distinct copies $G^i,G^j$ share two vertices $a,b$, then $\{a,b\}$ is neither an edge of $G^i$, nor $G^j$.

A G-packing satisfying (ii) is called an induced G-packing (see, e.g. [19]). Note that a weaker version of the above lemma, which only considered the existence of a large G-packing, regardless of the additional properties (i) and (ii), was used in [25] (see Theorem 5 of [25]) to prove the lower bound of (3). It is easy to see that Lemma 4 is near-optimal in the sense that the maximum size of any G-packing in $K_n$ cannot exceed $\left(\begin{array}{c}n\\ 2\end{array}\right)/e$.

We call the graph G in Lemma 4 the component graph, as it forms the basic component in the approximate decomposition. Following Theorem 3 it is natural to call a ${\mathcal{G}}_r(3r-4,3)$-free r-graph $\mathcal{H}\subseteq \left(\begin{array}{c}[n]\\ r\end{array}\right)$ optimal if it has roughly $(\frac{1}{r^2-r-1}+o(1))n^2$ edges as $n\to \infty$.

The following construction summarizes the main steps taken to prove the lower bound in Theorem 3.

Construction 5

Given $\mathcal{H}$, an optimal ${\mathcal{G}}_r(3r-4,3)$-free r-graph, we construct an optimal ${\mathcal{G}}_{r+1}(3(r+1)-4,3)$-free $(r+1)$-graph by performing the following three steps.

• By applying Lemma 4 with a carefully designed component graph $G_t$ (see Subsection 4.1), we approximately decompose the complete graph $K_n$ to $l=(1-ϵ)n^2/2|G_t|$ edge disjoint copies of $G_t$, say, $G_t^1,G_t^2,\dots ,G_t^l$;

• For $1\le i\le l$, by embedding in $V(G_t^i)$ many copies of $\mathcal{H}$ in a suitable way (see Subsection 4.2) we get an $(r+1)$-graph $G_t^i(\mathcal{H})$ (see Lemma 16);

• Output the $(r+1)$-graph $\mathcal{F}:={\cup }_{i=1}^l{G}_t^i(\mathcal{H})$, the edge disjoint union of the $G_t^i(\mathcal{H})$'s (see Subsection 4.3).

The base case, i.e., the optimal ${\mathcal{G}}_r(3r-4,3)$-free r-graph for $r=3$ was given by Glock [25]. Then, by applying Construction 5 iteratively, one can construct optimal ${\mathcal{G}}_r(3r-4,3)$-free r-graphs for all $r\ge 3$. The reader is referred to Section 4 for more details.

In the beginning of the last subsection it was mentioned that an $(n,r,2)$-packing is also a ${\mathcal{G}}_r(3r-4,3)$-free r-graph. However, this is not true in general, namely for $r>k\ge 3$, an $(n,r,k)$-packing is not necessarily a ${\mathcal{G}}_r(3r-2k,3)$-free r-graph, as $3(k-1)<2k$ if and only if $k<3$.

Our next result provides new lower and upper bounds for $\pi (r,k,3)$ for any fixed $r>k\ge 2$.

Theorem 6

For any fixed integers $r>k\ge 2$,

$$\frac{1}{r^k-r}\le \underset{n\to \infty }{\lim \inf }\frac{f_r(n,3r-2k,3)}{n^k}\le \underset{n\to \infty }{\lim \sup }\frac{f_r(n,3r-2k,3)}{n^k}\le \frac{1}{k!\left(\begin{array}{c}r\\ k\end{array}\right)-\frac{k!}{2}}.$$

One can easily check that for r much larger than k the gap between the lower and upper bounds in Theorem 6 is quite small. For example, let $r=\frac{k!}{2}$ and k be sufficiently large, then the two bounds almost match, as $r^k\approx k!\left(\begin{array}{c}r\\ k\end{array}\right)$. On the contrary, if r is approximately k, the lower bound becomes even weaker than that of (4). We omit the detailed computation.

The upper bound in Theorem 6, which includes that of Theorem 3 as a special case, follows from a weighted counting argument, and is presented in Section 3. The lower bound is proved by an algebraic construction, which relies on a new matrix property called strongly 3-perfect hashing, which is introduced below in Definition 20. The following lemma shows that in order to construct a ${\mathcal{G}}_r(3r-2k,3)$-free r-graph it is sufficient to construct a matrix with this property.

Lemma 7

Let $r>k\ge 2,\mathit{\text{and}}q$ be integers. If $\mathcal{M}$ is a strongly 3-perfect hashing q-ary matrix of order $r\times q^k$, then it induces a ${\mathcal{G}}_r(3r-2k,3)$-free r-partite r-graph ${\mathcal{H}}_{\mathcal{M}}$ over $n=rq$ vertices and $q^k$ edges, where the vertices can be partitioned to r disjoint parts $V_1,\dots ,V_r$ of size q each.

The proof of Lemma 7 is given in Subsection 5.1. Indeed, the multipartite r-graph constructed using Lemma 7 is optimal up to a constant, in the sense that it is easy to verify by the pigeonhole principle that any ${\mathcal{G}}_r(3r-2k,3)$-free r-partite r-graph, which has equal part size q, can have at most $2q^k$ edges.

The next construction outlines the main ingredients in proving the lower bound of Theorem 6.

Construction 8 Construction proving the lower bound of Theorem 6

By induction we assume that $f_r(n,3r-2k,3)\ge \frac{n^k}{r^k-r}-a{n}^{k-1}$ holds for every integer less than n, where $a=a(r,k)$ is some constant not depending on n, and we prove the statement for n.

• For fixed $r,k$, let q be the largest prime power satisfying $rq\le n$. By using the algebraic construction given in Subsections 5.2 and 5.3 we obtain an $r\times q^k$ q-ary strongly 3-perfect hashing matrix $\mathcal{M}$, which by Lemma 7 induces an r-partite r-graph ${\mathcal{H}}_{\mathcal{M}}$ over r vertex parts $V_1,\dots ,V_r$;

• By the induction hypothesis construct on each vertex part $V_i$ a ${\mathcal{G}}_r(3r-2k,3)$-free r-graph ${\mathcal{H}}_i$ with at least $\frac{q^k}{r^k-r}-a{q}^{k-1}$ edges;

• Output the r-graph $\mathcal{F}:=({\cup }_{i=1}^r{\mathcal{H}}_i)\cup {\mathcal{H}}_{\mathcal{M}}$, whose edges are the disjoint union of the edges of ${\mathcal{H}}_i,1\le i\le r$ and ${\mathcal{H}}_{\mathcal{M}}$.

The r-graph $\mathcal{F}$ has rq vertices and at least

$$q^k+r\cdot (\frac{q^k}{r^k-r}-a{q}^{k-1})=\frac{{(rq)}^k}{r^k-r}-ar{q}^{k-1}$$

edges. In order to complete the induction step it remains to show that $\mathcal{F}$ is ${\mathcal{G}}_r(3r-2k,3)$-free, and that the number of its edges is at least $\frac{n^k}{r^k-r}-a{n}^{k-1}$. The detailed proof is given in Section 5.

The rest of the paper is organized as follows. In Section 2 we briefly introduce two combinatorial problems which are closely related to the study of ${\pi }_d(r,k,e)$. In Section 3 we present the proof of the upper bound stated in Theorem 6. In Sections 4 and 5 we present the proofs of the lower bounds stated in Theorem 3, Theorem 6, respectively.