On asymptotic local Tur\'an problems

TL;DR

This paper investigates the asymptotic behavior of the minimum edge density in hypergraphs with the $(q,p)$-property, providing new results that confirm some conjectures and counterexamples for others.

Contribution

It offers partial positive answers to a conjecture about asymptotic densities and introduces new constructions that challenge existing beliefs for certain parameter ranges.

Findings

Confirmed the conjecture for a small range of real numbers.

Provided counterexamples for many other parameter ranges.

Advanced understanding of hypergraph density thresholds.

Abstract

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and let $t_r(q,p)=\lim_{n\to\infty}t_r(n,q,p)$. A disjoint union of $k$ complete hypergraphs has $(q,\lceil q/k\rceil)$-property, which gives $t_r((q,\lceil{q/k}\rceil))\le 1/k^{r-1}$. The first author, Huang and R\"odl showed that these constructions are the best asymptotically, that is, $\lim_{q\to\infty}t_r((q,\lceil{q/k}\rceil))=1/k^{r-1}$. They asked whether it is true for all real number $\gamma\ge1$ that $\lim_{q\to\infty}t_r((q,\lceil{q/\gamma}\rceil))=1/\lfloor{\gamma}\rfloor^{r-1}$. In this paper, we give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges.

Equations

$$t_r(q,p)=\underset{n\to \infty }{\lim }\frac{T_r(n,q,p)}{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}$$

$$t_2(q,p)=1/\lfloor \frac{q-1}{p-1}\rfloor .$$

$$t_r(q,p)=1\text{if}q\le \frac{r}{r-1}(p-1).$$

$${\hat{t}}_r(\gamma )=\underset{q\to \infty }{\lim }{t}_r(q,\lceil q/\gamma \rceil ).$$

$${\hat{t}}_2(\gamma )=1/\lfloor \gamma \rfloor$$

$${\hat{t}}_r(\gamma )=1$$

$${\hat{t}}_r(k)=\frac{1}{k^{r-1}}.$$

$$t_3(2p+1,p+1)=\frac{1}{4}.$$

$${\hat{t}}_r(\gamma )=1/\lfloor \gamma {\rfloor }^{r-1}.$$

$${\hat{t}}_r(\gamma )\ge \frac{1}{{\gamma }^{r-1}}$$

$${\hat{t}}_r(\gamma )\ge t_r(q,\lceil q/\gamma \rceil ).$$

$${\hat{t}}_r(\gamma )={\hat{t}}_r(\alpha ).$$

$${\hat{t}}_3(\gamma )=\frac{1}{4}.$$

$$t_3(k+1,3)=\frac{4}{k^2}.$$

$${\hat{t}}_3(\gamma )\ge \frac{4}{k^2}.$$

$${\hat{t}}_3(\gamma )=\frac{1}{t^2}.$$

$${\hat{t}}_r(\gamma )\le {\left(k-1+(2m+1){\left((m+1{)}^r-m^r\right)}^{-\frac{1}{r-1}}\right)}^{-r+1}$$

$${\hat{t}}_r(\gamma )\le {\left(k-1+{\left(\sum _{i=1}^m(-1{)}^{i+1}(\genfrac{}{}{0ex}{}{m}{i}){\left(1-\frac{i}{m}\right)}^r\right)}^{-\frac{1}{r-1}}\right)}^{-r+1}$$

$$|\mathcal{F}|\ge \frac{s}{t}\left(\genfrac{}{}{0ex}{}{tl}{r}\right).$$

$$|\mathcal{F}|\ge \sum _{i=1}^{sl}f(i).$$

$$\sum _{i=1}^{sl}f(i)=a\sum _{j=1}^{tl}\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right)+\sum _{j=1}^{tl}\delta (j)\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right)=a\left(\genfrac{}{}{0ex}{}{tl}{r}\right)+\sum _{j=1}^{tl}\delta (j)\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right).$$

$$\sum _{j=1}^{tl}\delta (j)\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right)\ge \frac{b}{t}\left(\genfrac{}{}{0ex}{}{tl}{r}\right)=\frac{b}{t}\sum _{j=1}^{tl}\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right).$$

$$\sum _{j=tl-k+1}^{tl}\left(\delta (j)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right)\ge \sum _{j=tl-k+1}^{tl}\left(\delta (j)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k}{r-1}\right).$$

$$\sum _{j=tl-k+1}^{tl}\delta (j)=\lceil \frac{ks}{t}\rceil -ka.$$

$$\sum _{j=tl-k}^{tl}\left(\delta (j)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{j-1}{r-1}\right)$$

$$=\left(\lceil \frac{ks}{t}\rceil -\frac{ks}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k}{r-1}\right)+\left(\delta (tl-k)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k-1}{r-1}\right)$$

$$\ge \left(\lceil \frac{ks}{t}\rceil -\frac{ks}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k-1}{r-1}\right)+\left(\delta (tl-k)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k-1}{r-1}\right)$$

$$=\sum _{j=tl-k}^{tl}\left(\delta (j)-\frac{b}{t}\right)\left(\genfrac{}{}{0ex}{}{tl-k-1}{r-1}\right).$$

$$\frac{s\left(\genfrac{}{}{0ex}{}{tl}{r}\right)}{t\left(\genfrac{}{}{0ex}{}{sl}{r}\right)}>(1-ϵ/2){\left(\frac{t}{s}\right)}^{r-1}.$$

$$\frac{|{\mathcal{H}}_0|}{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}\ge (1-ϵ/2)\frac{|{\mathcal{H}}_i|}{\left(\genfrac{}{}{0ex}{}{|X_i|}{r}\right)}\ge (1-ϵ/2)\frac{s\left(\genfrac{}{}{0ex}{}{tl}{r}\right)}{t\left(\genfrac{}{}{0ex}{}{sl}{r}\right)}\ge (1-ϵ/2{)}^2{\left(\frac{t}{s}\right)}^{r-1}.$$

$$t_r(q_1+q_2,p_1+p_2-1)\ge \min \{t_r(q_1,p_1),t_r(q_2,p_2)\}$$

$$\min \{t_r(n,q_1,p_1),t_r(n,q_2,p_2)\}\ge (1-ϵ)x,$$

Full text

1 Introduction

A hypergraph $\mathcal{H}$ is a collection of subsets of a finite set $V(\mathcal{H})$. Members of $V(\mathcal{H})$ are called vertices of $\mathcal{H}$, and members of $\mathcal{H}$ are called edges. An $r$-uniform hypergraph (or $r$-graph for short) is a hypergraph whose edges all have size $r$. For integers $q\geq p\geq r\geq 2$, an $r$-uniform hypergraph $H$ has $(q,p)$-property if for any $A\in\binom{V(\mathcal{H})}{q}$ there exists $B\in\binom{A}{p}$ such that $\binom{B}{r}\subset\mathcal{H}$. Let $T_{r}(n,q,p)$ be the minimum integer such that an $n$-vertex $r$-graph with $(q,p)$-property has $T_{r}(n,q,p)$ edges. By a simple averaging argument, Katona, Nemetz and Simonovits [6] showed that $T_{r}(n,q,p)/\binom{n}{r}$ is monotone increasing. Therefore, the limits

[TABLE]

exist. For the graph case, Erdös and Spencer [1] showed that

[TABLE]

For general $r$, the first author and Stechkin [5] proved that

[TABLE]

In this paper, we focus on the asymptotic behavior of $t_{r}(q,p)$.

Definition 1.1**.**

For every real number $\gamma\geq 1$, let

[TABLE]

See the Appendix for the proof of the existence of the limits.

Clearly, $\hat{t}_{r}(\gamma)$ is monotone non-increasing. Equation (1) implies

[TABLE]

for every $\gamma\geq 1$, and equation (2) implies

[TABLE]

for every $r\geq 2$ and $1\leq\gamma<\frac{r}{r-1}$. In [3] it is proved that $\hat{t}_{3}(2)=\frac{1}{4}$. This was recently generalized by the first author, Huang and Rödl [4] who showed the following:

Theorem 1.2** ([4]).**

For all integers $r\geq 2$, $k\geq 1$,

[TABLE]

For every integers $p\geq 3$,

[TABLE]

They asked whether it is true that for every real number $\gamma\geq 1$

[TABLE]

Towards (6), we first prove the following general lower bounds:

Theorem 1.3**.**

For every integer $r\geq 2$, and real number $\gamma\geq 1$,

[TABLE]

The next theorem, together with some known results, gives better lower bounds confirming (6) for a small range of real numbers.

Theorem 1.4**.**

For integers $q\geq r\geq 2$, and real number $\gamma\geq 1$,

[TABLE]

As a result, if $\hat{t}_{r}(\alpha)=t_{r}(q,\lceil q/\alpha\rceil)$ for some $\alpha\geq 1$, then for every real number $\gamma$ with $\alpha\leq\gamma<\frac{q}{\lceil q/\alpha\rceil-1}$,

[TABLE]

Remark**.**

Since $t_{2}(q,2)=\hat{t}_{2}(q-1)=\frac{1}{q-1}$ and $t_{r}(r,r)=\hat{t}_{r}(1)=1$, Theorem 1.4 implies (3) and (4).

Also, since $t_{3}(7,4)=\hat{t}_{3}(2)=1/4$ by Theorem 1.2, we have the following corollary:

Corollary 1.5**.**

For $2\leq\gamma<7/3$,

[TABLE]

Recently Fang, Gao, Ma and Song [2] have also proved (7).

Turán [7] proposed the following notorious conjectures:

Conjecture 1.6** ([7]).**

For every integers $k\geq 2$,

[TABLE]

This conjecture, if true, implies the following corollary:

Corollary 1.7**.**

If Conjecture 1.6 is true, then for any integer $k\geq 2$ and $\gamma<\frac{k+1}{2}$,

[TABLE]

In particular, for every integer $t\geq 1$ and $t\leq\gamma<t+\frac{1}{2}$, since $\hat{t}_{3}(t)=t_{3}(2t+1,3)=1/t^{2}$,

[TABLE]

On the other hand, we find new constructions with edge density smaller than that of the disjoint union of cliques, which give negative answers to (6) for many ranges of real numbers.

Theorem 1.8**.**

Let integers $r\geq 3$, $k\geq 1$.

• (1)

For every integer $m\geq 1$, and real number $k+\frac{m}{m+1}\leq\gamma<k+\frac{m+1}{m+2}$,

[TABLE]

• (2)

For every integer $3\leq m\leq r$, and real number $k+\frac{1}{m-1}\leq\gamma<k+\frac{1}{m-2}$,

[TABLE]

Recently, Fang, Gao, Ma and Song [2] also proved the upper bounds (9) for $k=1$ using the same construction. Moreover, they find constructions that give upper bounds better than (8) when $r=3$ and $m\geq 2$.

The rest of this paper is organized as follows:

In Section 2 we generalize ideas from [4] to prove the general lower bounds, that is, Theorem 1.3.

In Section 3, we proved Theorem 1.4 which, together with some known results, improves the lower bounds of Theorem 1.3.

In Section 4, we develop a framework that generates constructions for upper bounds and then use it to prove Theorem 1.8.

2 General lower bounds

In this section, we prove Theorem 1.3. A $(w,v)$-hole of an $r$-graph $\mathcal{H}$ is a set of vertices $W\subset V(\mathcal{H})$ such that $|W|=w$ and $v$ is the clique number, that is, the maximum integer such that there exists $A\in\binom{W}{v}$ such that $\binom{A}{r}\subset\mathcal{H}$. For any $w\geq u>v$, the existence of a $(w,v)$-hole indicates that the hypergraph does not have $(w,u)$-property.

Lemma 2.1** (Lemma 2.1 in [4]).**

For integers $q>p\geq r$, let $\mathcal{H}$ be an $r$-graph with $(q,p)$-property. If $\mathcal{H}$ has a $(w,v)$-hole $W$ such that $w\leq q-r$, then the induced sub-$r$-graph of $\mathcal{H}$ on $V(\mathcal{H})-W$, $\mathcal{H}\cap\binom{V(\mathcal{H})-W}{r}$, has $(q-w,p-v)$-property.

Since $\hat{t}_{r}(\gamma)$ is non-increasing, it suffices to show the lower bounds for rational numbers $\gamma=s/t$. We make use of the following lemma:

Lemma 2.2** (Generalization of Lemma 2.2 in [4]).**

For integers $s>t$, $r\geq 2$, and $l\geq 1$. Suppose an $r$-graph $\mathcal{F}$ on $sl$ vertices has $(q,p)$-property for all pairs $(q,p)$ with $q\leq sl$ and $p=\lceil tq/s\rceil$. Then

[TABLE]

Proof.

Let $g(i):=\lceil ti/s\rceil$ and let $f(i):=\binom{g(i)-1}{r-1}$. We claim that any induced sub-$r$-graph of $\mathcal{F}$ on $i$ vertices must contain a vertex with degree at least $f(i)$. This is because $\mathcal{F}$ has $(i,g(i))$-property. Hence any induced sub-$r$-graph of $\mathcal{F}$ on $i$ vertices contains a clique of size $g(i)$, and hence contains a vertex with degree at least $\binom{g(i)-1}{r-1}=f(i)$. Therefore,

[TABLE]

Suppose $s=at+b$ for some integers $a$ and $0\leq b<t$. It is not hard to check that $|g^{-1}(j)|=a$ or $a+1$. Let $\delta(j):=|g^{-1}(j)|-a$. So we have

[TABLE]

Now it suffices to show

[TABLE]

We prove the following stronger statement.

Claim**.**

For integer $1\leq k\leq tl$,

[TABLE]

Proof.

When $k=1$, the inequality is true. Suppose the inequality holds for $1\leq k<tl$. Note that for $1\leq k\leq tl$,

[TABLE]

We have

[TABLE]

This completes the proof of the claim. ∎

The lemma follows immediately by putting $k=tl$ in the claim.

∎

We generalize the notion of excess from [4]. For a pair $(q,p)$ with $tq\leq sp$, we call $e(q,p)=sp-tq$ the excess of the pair $(q,p)$. Note that since $q\geq p$, we always have $e(q,p)\leq(s-t)q$. In the following proof, a $(w,v)$-hole always has $sv<tw$ and $v\geq r-1$.

Proof of Theorem 1.3.

Given $\epsilon>0$, fix $l$ such that

[TABLE]

Then fix a much larger integer $L\geq 3t^{2}l^{2}$. Let $\mathcal{H}_{0}$ be an $r$-graph on $n$ vertices with $(sL,tL)$-property, where $n$ is sufficiently large. Our goal is to apply Lemma 2.2 on a subset $X\subset[n]$ with $\binom{|X|}{r}\geq(1-\epsilon/2)\binom{n}{r}$. This requires us to find such an $X$ such that $\binom{X}{r}\cap\mathcal{H}_{0}$ has no $(w,v)$-hole with $w\leq sl$ and $v<tw/s$.

We start with $\mathcal{H}_{0}$ and define $\mathcal{H}_{i}$ inductively. Let $q_{0}=sL$ and $p_{0}=tL$, $X_{0}=[n]$. Suppose $q_{i}\geq 2sl$ and $\mathcal{H}_{i}=\binom{X_{i}}{r}\cap\mathcal{H}_{0}$ has a $(w,v)$-hole $Y$ such that $w\leq sl$. Let $q_{i+1}=q_{i}-w$, $p_{i+1}=p_{i}-v$, and let $X_{i+1}=X_{i}\setminus{Y}$, $\mathcal{H}_{i+1}=\binom{X_{i+1}}{r}\cap\mathcal{H}_{0}$. Note that $e(q_{i+1},p_{i+1})=s(p_{i}-v)-t(q_{i}-w)\geq e(q_{i},p_{i})+1$. This implies that $i\leq e(q_{i},p_{i})\leq(s-t)q_{i}$.

The process keeps going unless, at some step $i$, $q_{i}<2sl$ or $\mathcal{H}_{i}$ contains no $(w,v)$-holes with $w\leq sl$.

If, at step $i$, $q_{i}<2sl$, then we have $sL=q_{0}\leq q_{i}+isl\leq q_{i}+(s-t)q_{i}sl<3s^{3}l$. This contradicts $L\geq 3s^{2}l^{2}$.

Hence, at some step $i$, we obtain $\mathcal{H}_{i}$ containing no $(w,v)$-holes with $w\leq sl$ such that $|X_{i}|\geq n-isl\geq n-(s-t)q_{0}sl$. For $n$ sufficiently large, we have $\binom{|X_{i}|}{r}\geq(1-\epsilon/2)\binom{n}{r}$. By Lemma 2.2,

[TABLE]

Let $n\to\infty$ and then let $\epsilon\to 0$. This completes the proof. ∎

3 Some better lower bounds

In this section, we prove Theorem 1.4. We make use of the following lemma:

Lemma 3.1**.**

For any integers $r\geq 2$, $q_{1}\geq p_{1}\geq r$, $q_{2}\geq p_{2}\geq r$,

[TABLE]

Proof.

Let $q_{0}=q_{1}+q_{2}$, $p_{0}=p_{1}+p_{2}-1$ and let $x=\min\{t_{r}(q_{1},p_{1}),~{}t_{r}(q_{2},p_{2})\}$. For any $\epsilon>0$, let $n_{0}$ be sufficiently large such that for any $n\geq n_{0}$,

[TABLE]

and

[TABLE]

For any $n\geq n_{0}+q_{1}$, let $G$ be an $r$-graph on $n$ vertices with $(q_{0},p_{0})$-property. If $G$ has $(q_{1},p_{1})$-property, then we have

[TABLE]

Otherwise, $\mathcal{G}$ has a $(q_{1},u)$-hole $Y$ where $u\leq p_{1}-1$. Let $\mathcal{G}^{\prime}=\binom{V(\mathcal{G})\setminus Y}{r}\cap G$. By Lemma 2.1, $\mathcal{G}^{\prime}$ has $(q_{2},p_{2})$-property. Therefore,

[TABLE]

Let $n\to\infty$ and let $\epsilon\to 0$, we infer $t_{r}(q_{0},p_{0})\geq x$. ∎

Lemma 3.2**.**

For any integers $q\geq p\geq r\geq 2$, $k\geq 0$,

[TABLE]

Proof.

We use induction on $k$. When $k=0$, equality holds. When $k\geq 1$, by Lemma 3.1 and the inductive hypothesis

[TABLE]

∎

Proof of Theorem 1.4.

Let $p=\lceil q/\gamma\rceil$, then

[TABLE]

Hence there exists $k_{0}$ such that, for any $k\geq k_{0}$, $(kq+q)/\gamma\geq k(p-1)+p$. Therefore, by Lemma 3.2

[TABLE]

Now suppose that $\hat{t}_{r}(\alpha)=t_{r}(q,\lceil q/\alpha\rceil)$. For any $\alpha\leq\gamma<\frac{q}{\lceil q/\alpha\rceil-1}$, we have $\lceil q/\alpha\rceil-1<\frac{q}{\gamma}\leq\frac{q}{\alpha}$, hence $\lceil q/\gamma\rceil=\lceil q/\alpha\rceil$, $\hat{t}_{r}(\gamma)\geq t_{r}(q,\lceil q/\gamma\rceil)=t_{r}(q,\lceil q/\alpha\rceil)=\hat{t}_{r}(\alpha)$. On the other hand, since the function $\hat{t}_{r}$ is non-increasing, we have $\hat{t}_{r}(\gamma)\leq\hat{t}_{r}(\alpha)$. Therefore, $\hat{t}_{r}(\gamma)=t_{r}(q,p)$. ∎

Corollary 1.5 and Corollary 1.7 follow easily from Theorem 1.4.

4 Constructions for upper bounds

In this section, we describe a framework that generates constructions for upper bounds of $\hat{t}_{r}(\gamma)$ and then use it to prove Theorem 1.8. We will also briefly introduce the constructions used by Fang, Gao, Ma and Song [2].

Definition 4.1**.**

Fix a hypergraph $\mathcal{H}$ on $[t]=\{1,\dots,t\}$. Let $\mathcal{F}(\mathcal{H},r)$ be a family of $r$-graphs $\mathcal{G}$ with the following properties: vertices of $\mathcal{G}$ can be partitioned into $t$ disjoint sets $V_{1},\dots,V_{t}$ such that

[TABLE]

For example, a 3-graph $\mathcal{G}\in\mathcal{F}(C_{3},3)$ consists of all triples $e$ such that $|e\cap V_{i}|\geq 2$ for some $i\in[3]$. Let

[TABLE]

and let

[TABLE]

By an averaging argument, we know that, when $\mathcal{H}$ and $r$ are fixed, $\rho(\mathcal{H},r,n)$ is non-decreasing. Hence, the limit $\rho(\mathcal{H},r)$ exists.

Next, we introduce two parameters of a hypergraph $\mathcal{H}$ that will make it more convenient to describe properties of the family $\mathcal{F}(\mathcal{H},r)$. For every integer $t\geq 1$, let

[TABLE]

Given a hypergraph $\mathcal{H}$ on $[t]$, we define a function $f_{\mathcal{H}}$ on $\Delta_{t-1}$,

[TABLE]

Definition 4.2**.**

Given a hypergraph $\mathcal{H}$ on $[t]$, let

[TABLE]

Since $f_{\mathcal{H}}$ is continuous and $\Delta_{t-1}$ is compact, $f(\mathcal{H})$ is well-defined. The intuition is, given a vertex set $A$ in a hypergraph $\mathcal{H}\in\mathcal{F}(\mathcal{H},r)$, we can let $(x_{1},\dots,x_{t})$ be the vertex distribution of $A$ among different parts of $\mathcal{H}$. Now if $f_{\mathcal{H}}$ is large, then no matter how the vertices in $A$ are distributed, we can always find a clique in $A$ of size $f_{\mathcal{H}}|A|$. The next lemma reveals the relation between $f_{\mathcal{H}}$ and the local Turán property of a hypergraph in $\rho(\mathcal{H},r,n)$ in a more formal way.

Lemma 4.3**.**

Let $\mathcal{H}$ be a hypergraph on $[t]$ and let $\gamma=1/f(\mathcal{H})$. Then $\forall q\geq r\geq 2$ and $\forall\mathcal{G}\in\mathcal{F}(\mathcal{H},r)$, $\mathcal{G}$ has $(q,\lceil q/\gamma\rceil)$-property. Therefore,

[TABLE]

Proof.

Let $V_{1},\dots,V_{t}$ be disjoint sets of vertices of $\mathcal{G}$ satisfying (10). Let $W$ be a set of $q$ vertices of $\mathcal{G}$. Let $a_{i}=|W\cap V_{i}|$, $\forall 1\leq i\leq t$. Clearly, $(a_{1}/q,\dots,a_{t}/q)$ is a vector in $\Delta_{t-1}$. Hence by definition, there exists an edge $e\in\mathcal{H}$ such that

[TABLE]

Since $\sum_{i\in e}a_{i}$ is an integer, we have $\sum_{i\in e}a_{i}\geq\lceil q/\gamma\rceil$. By equation (10), $W\cap(\cup_{i\in e}V_{i})$ spans a clique of size $\sum_{i\in e}a_{i}\geq\lceil q/\gamma\rceil$ in $\mathcal{G}$. Therefore, $\mathcal{G}$ has $(q,\lceil q/\gamma\rceil)$-property.

As a result, we have $t_{r}(q,\lceil q/\gamma\rceil,n)\leq\rho(\mathcal{H},r,n)$. Letting $n\to\infty$ and $q\to\infty$ gives $\hat{t}_{r}(\gamma)\leq\rho(\mathcal{H},r)$. ∎

A hypergraph $\mathcal{H}$ is said to be the disjoint union of $k$ hypergraphs $\mathcal{H}_{1},\dots,\mathcal{H}_{k}$ where $k\geq 2$ if $\mathcal{H}_{1},\dots,\mathcal{H}_{k}$ have disjoint vertex sets and $\mathcal{H}=\mathcal{H}_{1}\cup\dots\cup\mathcal{H}_{k}$.

Lemma 4.4**.**

Let $k\geq 2$ and let $\mathcal{H}$ be the disjoint union of $k$ hypergraphs $\mathcal{H}_{1},\dots,\mathcal{H}_{k}$, then

[TABLE]

Proof.

It suffices to show this for $k=2$. When $k=2$, by definition,

[TABLE]

Clearly, the minimum is obtained when $cf(\mathcal{H}_{1})=(1-c)f(\mathcal{H}_{2})$, which implies

[TABLE]

Hence

[TABLE]

∎

Given a hypergraph $\mathcal{H}$ on $[t]$ and $r\geq 2$, we define a function $g_{\mathcal{H},r}$ on $\Delta_{t-1}$,

[TABLE]

where $\sigma_{\mathcal{H}}(y_{1},\dots,y_{r})=1$ if there exists $e\in\mathcal{H}$ such that $y_{i}\in e,~{}\forall 1\leq i\leq r$; otherwise, $\sigma(y_{1},\dots,y_{r})=0$.

Definition 4.5**.**

Given a hypergraph $\mathcal{H}$ on $[t]$ and an integer $r\geq 2$, let

[TABLE]

Since $g_{\mathcal{H},r}$ is continuous and $\Delta_{t-1}$ is compact, $g(\mathcal{H},r)$ is well-defined.

Lemma 4.6**.**

Let $r,k\geq 2$ and let $\mathcal{H}$ be the disjoint union of $k$ hypergraphs $\mathcal{H}_{1},\dots,\mathcal{H}_{k}$, then

[TABLE]

Proof.

Again, it is sufficient to show the statement for $k=2$. When $k=2$, by definition,

[TABLE]

Let $g(\mathcal{H}_{1},r)=g_{1}$, $g(\mathcal{H}_{2},r)=g_{2}$ and let $b=(g_{1}/g_{2})^{\frac{1}{r-1}}$. Applying Jensen’s inequality to the convex function $x^{r}$,

[TABLE]

Equality holds if and only if $c=(1-c)/b$ which is equivalent to $c=1/(1+b)$. Hence this minimum is obtainable. Therefore,

[TABLE]

∎

Lemma 4.7**.**

Let $\mathcal{H}$ be a hypergraph on $[t]$ and $r\geq 2$. Then

[TABLE]

Proof.

For any $\mathcal{G}\in\mathcal{F}(\mathcal{H},r)$, let $V_{1},\dots,V_{t}$ be disjoint set of vertices of $\mathcal{G}$ satisfying (10). Given $y=(y_{1},\dots,y_{r})\in[t]^{r}$, $\forall 1\leq i\leq t$, let $m_{y}(i)$ denote the number of entries equal to $i$ in $y$. By (10), we can count the number of edges in $\mathcal{G}$ as following: for each $y=(y_{1},\dots,y_{r})\in[t]^{r}$, if $\sigma_{\mathcal{H}}(y)=1$, then the number of (ordered) edges $e=(v_{1},\dots,v_{r})$ such that $v_{i}\in V_{y_{i}}$ for all $1\leq i\leq r$ is exactly

[TABLE]

If we sum the quantity above over all $y\in[t]^{r}$, then each edge in $\mathcal{G}$ is counted exactly $r\,!$ times. Hence,

[TABLE]

where $c_{1}$ is a constant that depends only on $r$. This implies that

[TABLE]

Letting $n\to\infty$ gives $\rho(\mathcal{H},r)\geq g(\mathcal{H},r)$.

On the other hand, let $x=(x_{1},\dots,x_{t})\in\Delta_{t-1}$ be the vector such that $g_{\mathcal{H},r}(x)=g(\mathcal{H},r)$. We can pick a sequence of $r$-graphs $\{\mathcal{G}_{n}\}_{n\geq 1}\subset\mathcal{F}(\mathcal{H},r)$ such that $v(\mathcal{G}_{n})=n$ and $|V_{i}(\mathcal{G}_{n})|/n\to x_{i}$ as $n\to\infty$, $\forall 1\leq i\leq t$. By definition, there exists a constant $c_{2}$ that depends only on $r$ such that

[TABLE]

Letting $n\to\infty$ gives $g(\mathcal{H},r)\geq\rho(\mathcal{H},r)$.

∎

Definition 4.8**.**

For integers $t>m\geq 2$, an $m$-uniform tight cycle of length $t$, denoted by $C^{m}_{t}$ is an $m$-graph on $[t]$ with $t$ edges $e_{i}=\{i,~{}{i+1},~{}\dots,~{}{i+m-1}\}$, $1\leq i\leq t$ (vertices are modulo $t$).

We will use tight cycles of the forms $C^{m+1}_{2m+1}$ and $C^{m}_{m-1}$ as building blocks to give constructions that prove Theorem 1.8. Next, we introduce two lemmas that compute the parameters $f$ and $g$ of tight cycles.

Lemma 4.9**.**

For every inters $t>m\geq 2$,

[TABLE]

Proof.

We will use the notations in Definition 4.8. For any $x\in\Delta_{t-1}$, let $x=(x_{1},\dots,x_{t})$. Then

[TABLE]

Hence, $f(C^{m}_{t})\geq m/t$. On the other hand, if we let $x=(1/t,\dots,1/t)$, then $f_{C^{m}_{t}}(x)=m/t$. Therefore, $f(C^{m}_{t})=m/t$. ∎

Lemma 4.10**.**

Let $r\geq 3$ be an integer.

• (1)

For every integer $m\geq 1$,

[TABLE]

• (2)

For every integer $3\leq m\leq r$,

[TABLE]

Proof.

• (1)

Throughout this proof the indices in $[2m+1]$ are modulo $2m+1$. For any $x=(x_{1},\dots,x_{2m+1})\in X_{2m+1}$, we claim that

[TABLE]

For any $y=(y_{1},\dots,y_{r})\in[2m+1]^{r}$ such that $\sigma_{C^{m+1}_{2m+1}}(y)=1$, it suffices to show that the coefficient of the ordered monomial $x_{y_{1}}\dots x_{y_{r}}$ is $1$ in the right-hand side of (11). Let $j$ be the minimum integer such that there exists $a\in[2m+1]$ such that $y_{i}\in\{a,\dots,a+j\}$, $\forall 1\leq i\leq r$. By definition, such $j$ must exist and satisfy $0\leq j\leq m$. Then the ordered monomial $x_{y_{1}}\dots x_{y_{r}}$ is contained in $(x_{i}+\dots+x_{i+m})^{r}$ for $i\in\{a+j-m,\dots,a\}$, which contributes $m-j+1$ to its coefficient; and is contained in $(x_{i}+\dots+x_{i+m-1})^{r}$ for $i\in\{a+j-m+1,\dots,a\}$, which contributes $-(m-j)$ to its coefficient. Hence its coefficient is exactly 1. This proves the claim.

Now let $x=(1/(2m+1),\dots,1/(2m+1))$. We have

[TABLE]

• (2)

For $x=(x_{1},\dots,x_{m})\in X_{m}$, note that the polynomial $g_{C^{m-1}_{m},r}(x)$ is the sum of all ordered monomials of degree $r$ that is missing some $x_{i}$, each of them has coefficient 1. For any subset $S\in[m]$, observe that $(\sum_{i=1}^{m}x_{i}-\sum_{i\in S}x_{i})^{r}$ is the sum of all ordered monomials of degree $r$ that is missing all $x_{i}$ with $i\in S$, each of them has coefficient 1. Hence by the Inclusion-Exclusion Principle, we have

[TABLE]

Now let $x=(1/(2m+1),\dots,1/(2m+1))$. We have

[TABLE]

∎

Let $I_{k}$ be the $1$-uniform hypergraph with $k$ vertices and $k$ edges. Given a hypergraph $\mathcal{H}$, let $\mathcal{H}\cup I_{k}$ denote the disjoint union of $\mathcal{H}$ and $I_{k}$. It is easy to check that $f(I_{k})=1/k$ and $g(I_{k},r)=k^{-r+1}$.

Proof of Theorem 1.8.

By Lemma 4.9 and Lemma 4.4,

[TABLE]

1. (1)

By Lemma 4.3, Lemma 4.7 and the fact that $\hat{t}_{r}$ is non-increasing, when $\gamma\geq k+\frac{m}{m+1}=1/f(C^{m+1}_{2m+1}\cup I_{k-1})$,

[TABLE]

By Lemma 4.10 and Lemma 4.6,

[TABLE]

Therefore,

[TABLE] 2. (2)

By Lemma 4.3, Lemma 4.7 and the fact that $\hat{t}_{r}$ is non-increasing, when $\gamma\geq k+\frac{1}{m-1}=1/f(C^{m-1}_{m}\cup I_{k-1})$,

[TABLE]

By Lemma 4.10 and Lemma 4.6,

[TABLE]

Therefore,

[TABLE]

∎

Fang, Gao, Ma and Song [2] make use of the following hypergraphs.

Definition 4.11**.**

For integer $m\geq 2$, let $W_{m}$ be a hypergraph on $[m+1]$ defined as following:

[TABLE]

It is not hard to check that $f(W_{m})=m/(2m-1)$ and $g(W_{m},3)=(5m+4)/9m$. Hence, by Lemma 4.3, Lemma 4.4, Lemma 4.7 and Lemma 4.6, for every integers $k,m\geq 1$ and real number $\gamma\geq k+\frac{m}{m+1}$,

[TABLE]

This is strictly better than (8) when $r=3$ and $m\geq 2$.

5 Concluding remarks

• •

By Lemma 4.3 and Lemma 4.7 we know that

[TABLE]

Is this bound tight? All known correct constructions so far support this intuition. To begin with, can we show that $\mathcal{F}(C^{2}_{3},r)$ provides the best constructions for $\hat{t}_{r}(\frac{3}{2})$? In other words, is it true that

[TABLE]

The case $r=3$ has been confirmed recently by Fang, Gao, Ma and Song [2].

• •

For $0\leq\epsilon<1$, we know from the proof of Theorem 1.8 that $\hat{t}_{r}(k+\epsilon)\leq\left(k-1+\hat{t}_{r}(1+\epsilon)^{-\frac{1}{r-1}}\right)^{-r+1}$. Is this tight? In other words, is it true that the best construction for $\hat{t}_{r}(k+\epsilon)$ comes from taking the disjoint union of the best construction for $\hat{t}_{r}(1+\epsilon)$ and $(k-1)$ cliques? This is true for $\epsilon=0$ by Theorem 1.2.

Aknowledgement

We would like to thank Hao Huang, Jie Ma and Hehui Wu for helpful discussions.

Appendix A

Proposition 5.1**.**

For every integer $r\geq 2$ and real number $\gamma>1$, the limits

[TABLE]

exist.

Proof.

Let $T=\limsup_{q\to\infty}t_{r}(q,\lceil q/\gamma\rceil)$. Let $\epsilon>0$ be an arbitrarily small real number. We need to show that there exists $q_{0}=q_{0}(\epsilon)$ such that for all $q\geq q_{0}$, $t_{r}(q,\lceil q/\gamma\rceil)\geq T-\epsilon$. By definition, there exists $q^{\prime}$ such that $t_{r}(q^{\prime},p^{\prime})\geq T-\epsilon$ where $p^{\prime}=\lceil q^{\prime}/\gamma\rceil$. Clearly, $q^{\prime}/(p^{\prime}-1)>\gamma$. Let $\delta=\frac{1}{\gamma}-(p^{\prime}-1)/q^{\prime}$. Let $q_{0}$ be large enough such that for every $q\geq q_{0}$

[TABLE]

and

[TABLE]

Fix a $q\geq q_{0}$, and let $p=\lceil q/\gamma\rceil$. For $k\geq 1$, let $q_{k}=q-kq^{\prime}$, $p_{k}=p-k(p^{\prime}-1)$. By Lemma 3.1, as long as $t_{r}(q_{k},p_{k})$ is well defined,

[TABLE]

Let $k^{\prime}$ be the unique integer such that $0\leq q_{k^{\prime}}-p_{k^{\prime}}\leq q^{\prime}-p^{\prime}$. Not hard to check that

[TABLE]

By (13), we have

[TABLE]

Hence, by (14),

[TABLE]

Note that by definition, $t_{r}(a,b)\geq t_{r}(a-1,b-1)\geq t_{r}(a,b-1)$. So we have,

[TABLE]

Therefore,

[TABLE]

∎