Focal-free uniform hypergraphs and codes

Xinqi Huang School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China. Emails: [email protected], [email protected]. Chong Shangguan Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, 266237, China; Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, 266237, China. Email: [email protected]. Xiande Zhang School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China; Hefei National Laboratory, University of Science and Technology of China, Hefei, 230088, Anhui, China. Email: [email protected]. Yuhao Zhao¹¹footnotemark: 1

Abstract

Motivated by the study of a variant of sunflowers, Alon and Holzman recently introduced focal-free hypergraphs. In this paper, we show that there is an interesting connection between the maximum size of focal-free hypergraphs and the renowned Erdős Matching Conjecture on the maximum number of edges that can be contained in a uniform hypergraph with bounded matching number. As a consequence, we give asymptotically optimal bounds on the maximum sizes of focal-free uniform hypergraphs and codes, thereby significantly improving the previous results of Alon and Holzman. Moreover, by using the existentce results of combinatorial designs and orthogonal arrays, we are able to explicitly determine the exact sizes of maximum focal-free uniform hypergraphs and codes for a wide range of parameters.

1 Introduction

Questions asking for the maximum sizes of set systems without containing certain forbidden configurations are widely studied in extremal combinatorics. They usually have various applications in coding theory and theoretical computer science (see, e.g., [20]). In this paper, we study extremal problems concerning forbidding $r$ -focal hypergraphs, and present asymptotically optimal bounds on their maximum sizes.

We begin with some definitions. For a positive integer $n$ , write $[n]$ for the set $\{1,2,\ldots,n\}$ . For a finite set $X$ , let $2^{X}$ denote the power set of $X$ and $\binom{X}{k}$ denote the set of all $k$ -subsets of $X$ . A hypergraph $\mathcal{F}$ on the vertex set $V(\mathcal{F})$ is a family of distinct subsets (called edges) of $V(\mathcal{F})$ . We assume without loss of generality that every vertex in $V(\mathcal{F})$ is contained in at least one edge in $\mathcal{F}$ .

Definition 1.1.

A family of $r$ distinct sets $A_{0},A_{1},\ldots,A_{r-1}\subseteq 2^{[n]}$ is said to be an $r$ -focal hypergraph with focus $A_{0}$ if for every $x\in A_{0}$ , we have

\displaystyle|\{i\in[r-1]:x\in A_{i}\}|\geq r-2.

A hypergraph $\mathcal{F}\subseteq 2^{[n]}$ is $r$ -focal-free if it does not contain any $r$ -focal hypergraph.

Note that the above definition is trivial for $r=2$ ; therefore, we assume throughout that $r\geq 3$ . Focal-free hypergraphs were introduced by Alon and Holzman [1] when they were studying a variant of sunflowers, called near-sunflowers (see [1] and the end of this section for more backgrounds).

A hypergraph $\mathcal{F}\subseteq 2^{[n]}$ is said to be $k$ -uniform if $\mathcal{F}\subseteq\binom{[n]}{k}$ . Let $f_{r}(n,k)$ denote the maximum cardinality of an $n$ -vertex $r$ -focal-free $k$ -uniform hypergraph. Alon and Holzman (see [1, Theorem 5.2]) showed that¹¹1In [1], uniform focal-free hypergraphs were called “one-sided focal families”. for all $r\geq 3$ and $0\leq k\leq n$ ,

f_{r}(n,k)\leq(r-1)\cdot\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}\bigg{/}\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}.

(1)

They also commented that “a lower bound on $f_{r}(n,k)$ may be obtained by random choice with alterations, but optimizing the bound requires rather messy calculations.”

Quite surprisingly, we notice that packings provide a cheap lower bound for $f_{r}(n,k)$ . For integers $n>k>t\geq 2$ , an $(n,k,t)$ -packing is a $k$ -uniform hypergraph $\mathcal{P}\subseteq\binom{[n]}{k}$ such that $|A\cap B|<t$ for every two distinct $A,B\in\mathcal{P}$ . We observe that every $(n,k,\lceil\frac{(r-2)k}{r-1}\rceil)$ -packing $\mathcal{P}$ is $r$ -focal-free. Indeed, assume otherwise that there exist $A_{0},A_{1},\ldots,A_{r-1}\subseteq\mathcal{P}$ that form an $r$ -focal hypergraph with focus $A_{0}$ . Then, $A_{0}\setminus A_{1},\ldots,A_{0}\setminus A_{r-1}$ must be pairwise disjoint subsets of $A_{0}$ . Therefore, $\sum_{i=1}^{r-1}|A_{0}\setminus A_{i}|\leq|A_{0}|=k$ , which implies that there exists some $i\in[r-1]$ such that $|A_{0}\setminus A_{i}|\leq\lfloor\frac{k}{r-1}\rfloor$ . Consequently, $|A_{0}\cap A_{i}|\geq\lceil\frac{(r-2)k}{r-1}\rceil$ , a contradiction. A celebrated result of Rödl [26] showed that for fixed $k,t$ and sufficiently large $n$ , there exist asymptotically optimal $(n,k,t)$ -packings with cardinality $(1-o(1))\cdot\binom{n}{t}/\binom{k}{t}$ , where $o(1)\rightarrow 0$ as $n\rightarrow\infty$ . Together with the above discussion, it yields that

\displaystyle(1-o(1))\cdot\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}\bigg{/}\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}\leq f_{r}(n,k)\leq(r-1)\cdot\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}\bigg{/}\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}.

(2)

Note that for $k\geq r-1$ we have $\lceil\frac{(r-2)k}{r-1}\rceil<k$ , and determining $f_{r}(n,k)$ is a degenerate Turán-type extremal problem for hypergraphs. Conventionally speaking, if there exist two reals $\alpha_{r}(k)>0$ and $\beta_{r}(k)\in(0,k)$ such that $\lim_{n\rightarrow\infty}\frac{f_{r}(n,k)}{n^{\beta_{r}(k)}}=\alpha_{r}(k)$ , then we call $\alpha_{r}(k)$ and $\beta_{r}(k)$ the degenerate Turán density and Turán exponent of $f_{r}(n,k)$ , respectively. Clearly, (2) already shows that $\beta_{r}(k)=\lceil\frac{(r-2)k}{r-1}\rceil$ but it gives no hint on the existence or the exact value of $\alpha_{r}(k)$ .

Our first main result significantly improves upon (1) (and also on (2)) by explicitly determining the exact values of the degenerate Turán density and Turán exponent of $f_{r}(n,k)$ . In particular, to determine $\alpha_{r}(k)$ , we establish an interesting connection between focal-free uniform hypergraphs and the well-known Erdős Matching Conjecture [10]. Let $m(n,s,\lambda)$ denote the maximum number of edges that can be contained in an $s$ -uniform hypergraph on $n$ vertices that does not contain $\lambda$ pairwise disjoint edges. Erdős [10] famously conjectured that for all $n\geq s\lambda$ ,

m(n,s,\lambda)=\max\left\{\binom{n}{s}-\binom{n-\lambda+1}{s},\binom{s\lambda-1}{s}\right\}.

The above conjecture has been partially confirmed; see [16] and the references therein for details.

The following theorem shows that $\alpha_{r}(k)$ can be precisely determined in terms of $m(n,s,\lambda)$ .

Theorem 1.2.

For all fixed $r\geq 3$ and $k\geq 2$ , we have

\lim_{n\to\infty}f_{r}(n,k)\bigg{/}\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}=\frac{1}{\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}-m(k,\lfloor\frac{k}{r-1}\rfloor,\lambda)},

where $\lambda\in[r-1]$ is the unique integer that satisfies $k+\lambda\equiv 0\pmod{r-1}$ .

Note that the upper bound of Theorem 1.2, proved in Theorem 2.2, improves the upper bound in (1), since by a result of Frankl [14], $m(n,s,\lambda)\leq(\lambda-1)\binom{n-1}{s-1}$ .

Moreover, for $\lambda=1$ , it is clear that $m(k,\lfloor\frac{k}{r-1}\rfloor,1)=0$ ; in this case we can, in fact, prove $f_{r}(n,k)\leq\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}/\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}$ . An $(n,k,t)$ -packing $\mathcal{P}\subseteq\binom{[n]}{k}$ is said to be an $(n,k,t)$ -design if $|\mathcal{P}|=\binom{n}{t}/\binom{k}{t}$ . Combining with the known results on the existence of combinatorial designs (see [7, 21]), we are able to determine the exact values of $f_{r}(n,k)$ for infinitely many parameters.

Proposition 1.3.

Let $r\geq 3$ , $r-1\mid k+1$ , and $t=\lceil\frac{(r-2)k}{r-1}\rceil$ . Let $n\geq\max\{k,n_{0}\}$ , where $n_{0}=\binom{k}{k-t}t+t-1$ . If there exists an $(n,k,t)$ -design, then $f_{r}(n,k)=\binom{n}{t}/\binom{k}{t}$ . In particular, for every sufficiently large $n$ that satisfies $\binom{k-i}{t-i}\mid\binom{n-i}{t-i}$ for every $0\leq i\leq t-1$ , we have

f_{r}(n,k)=\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}\bigg{/}\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}.

As counterparts of focal-free uniform hypergraphs, focal-free codes are also of interest and were systematically studied by Alon and Holzman [1].

Definition 1.4.

Let $q\geq 2$ be an integer. A family $\bm{x}^{0},\bm{x}^{1},\dots,\bm{x}^{r-1}$ of $r$ distinct vectors in $[q]^{n}$ is said to be an $r$ -focal code with focus $\bm{x}^{0}$ if for every coordinate $i\in[n]$ , we have

\displaystyle|\{j\in[r-1]:x_{i}^{j}=x_{i}^{0}\}|\geq r-2.

A code $\mathcal{C}\subseteq[q]^{n}$ is $r$ -focal-free if it does not contain an $r$ -focal code as a subset.

Let $f_{r}^{q}(n)$ denote the maximum cardinality of an $r$ -focal-free code in $[q]^{n}$ . Alon and Holzman [1] proved that

c_{r}\left(\frac{q}{((q-1)(r-1)+1)^{1/(r-1)}}\right)^{n}\leq f_{r}^{q}(n)\leq(r-1)q^{\lceil\frac{(r-2)n}{r-1}\rceil},

(3)

where $c_{r}>0$ is a positive constant that depends only on $r$ . Moreover, for any prime power $q\geq n$ , they proved a better lower bound:

f_{r}^{q}(n)\geq q^{\lceil\frac{(r-2)n}{r-1}\rceil}.

(4)

Thus, the above results showed that for fixed $r,n$ and sufficiently large $q$ , we have

f_{r}^{q}(n)=\Theta_{n,r}(q^{\lceil\frac{(r-2)n}{r-1}\rceil}).

(5)

Therefore, it remains a natural problem to determine whether the limit $\lim_{q\to\infty}f_{r}^{q}(n)/q^{\lceil\frac{(r-2)n}{r-1}\rceil}$ exists.

Our second main result improves upon (5) by explicitly determining the value of the above limit.

Theorem 1.5.

For all fixed $r\geq 3$ and $n\geq 2$ , we have

\lim\limits_{q\to\infty}f_{r}^{q}(n)\big{/}q^{\lceil\frac{(r-2)n}{r-1}\rceil}=\frac{\binom{n}{\lceil\frac{(r-2)n}{r-1}\rceil}}{\binom{n}{\lceil\frac{(r-2)n}{r-1}\rceil}-m(n,\lfloor\frac{n}{r-1}\rfloor,\lambda)},

where $\lambda\in[r-1]$ is the unique integer that satisfies $n+\lambda\equiv 0\pmod{r-1}$ .

Similarly to Proposition 1.3, in the theorem below we obtain some exact results on $f_{r}^{q}(n)$ . Moreover, compared with Proposition 1.3, in Theorem 1.6, the requirements on the parameters are more flexible.

Theorem 1.6.

Let $q\geq r-1\geq 2$ and let $q=p_{1}^{e_{1}}\cdots p_{s}^{e_{s}}$ be the canonical integer factorization of $q\geq 2$ , where $p_{1},\ldots,p_{s}$ are distinct primes, $e_{1},\ldots,e_{s}$ are positive integers, and $p_{1}^{e_{1}}<\cdots<p_{s}^{e_{s}}$ . If $r-1\mid n+1$ and $2r-3<n\leq p_{1}^{e_{1}}+1$ , then

f_{r}^{q}(n)=q^{\lceil\frac{(r-2)n}{r-1}\rceil}.

We explain the main idea of our proofs here. The upper bounds on the limits in Theorems 1.2 and 1.5 are essentially proved by double counting (see Theorems 2.2 and 3.2 below). For $q\geq r-1$ and $r-1\mid n+1$ , we apply a slightly more sophisticated counting argument and prove a clean upper bound for $f_{r}^{q}(n)$ (without the lower order term, see Theorem 3.5), which eventually yields the exact result in Theorem 1.6.

The lower bounds on the limits in Theorems 1.2 and 1.5 are inspired by the aforementioned work of Rödl [26], and its further developments (see [15, 17, 22, 24]). In particular, the lower bound construction of Theorem 1.2 (see Theorem 2.5) is based on a powerful result of Frankl and Füredi [15] on the existence of near-optimal induced hypergraph packings; and the lower bound in Theorem 1.5 (see Theorem 3.4) applies a variant of Frankl and Füredi’s result, recently obtained by Liu and Shangguan [22], to induced packings in multi-partite hypergraphs, where the packings should respect the vertex partition of the host multi-partite hypergraph.

Outline of the paper.

For focal-free uniform hypergraphs, we will prove Theorem 1.2 in Section 2, where the upper and lower bounds of the limit are proved in Sections 2.1 and 2.2, respectively; furthermore, a short proof of Proposition 1.3 is also included in Section 2.1. For focal-free codes, since the proof of Theorem 1.5 is very similar to that of Theorem 1.2, we will sketch its proof in Appendix A; however, the proofs of Theorem 1.6 and Proposition 1.3 are quite different, and we will prove Theorem 1.6 in Section 3.2. Lastly, we will conclude this paper in Section 4 with some open questions.

Related work.

We will end this section by mentioning some related work. We remark that Alon and Holzman’s study of focal-free hypergraphs and codes was motivated by the study of near-sunflowers, which is a variant of the well-known combinatorial object sunflowers.

A family of $r$ distinct subsets of $[n]$ is an $r$ -near-sunflower if every $i\in[n]$ belongs to either $0,1,r-1$ or $r$ of the members in this family. It is easy to see that if $\mathcal{H}\subseteq 2^{[n]}$ is $r$ -near-sunflower-free then it is also $r$ -focal-free. Therefore, the $q=2$ case of (3) shows that every $r$ -near-sunflower-free family ${\mathcal{H}}\subseteq 2^{[n]}$ has cardinality at most $c^{n}$ , where $c<2$ is a constant depending only on $r$ . This in fact proved the Erdős–Szemerédi-type conjecture for near-sunflowers (which is weaker than the Erdős–Szemerédi conjecture for sunflowers). Alon and Holzman further posed an Erdős–Rado-type conjecture for near-sunflowers (which is again weaker than the Erdős–Rado conjecture for sunflowers), and it is still open. For more details on the Erdős–Rado and Erdős–Szemerédi conjectures for sunflowers, see, e.g. [2, 4, 9, 12, 13, 19, 23].

As illustrated in [1], focal-free hypergraphs and codes are also closely related to cover-free families [11, 15] and frameproof codes [5, 6, 22], which were extensively studied in combinatorics and coding theory. Indeed, when $r=3$ , $3$ -focal-free hypergraphs are equivalent to $2$ -cover-free families, and $3$ -focal-free codes are equivalent to $2$ -frameproof codes. The interested reader is referred to [1] for a more detailed discussion on the relation of focal-free hypergraphs and codes and various other combinatorial objects.

2 Focal-free hypergraphs

The goal of this section is to present the proof of Theorem 1.2. We will use the following definition. Let $\mathcal{F}\subseteq 2^{[n]}$ be a hypergraph and $A\in\mathcal{F}$ be an edge. A subset $T\subseteq A$ is called an own subset of $A$ (with respect to ${\mathcal{F}}$ ) if for every $B\in{\mathcal{F}}\setminus\{A\}$ , we have $T\nsubseteq B$ ; otherwise, $T$ is called a non-own subset of $A$ (with respect to ${\mathcal{F}}$ ).

The following observation presents sufficient and necessary conditions for the existence of an $r$ -focal hypergraph. It will be very useful in our proof of Theorem 1.2.

Observation 2.1.

Let $\mathcal{F}$ be a hypergraph with $|\mathcal{F}|\geq r$ . Then the following hold:

(i)

If $A\in\mathcal{F}$ admits a partition $A=T_{1}\cup\cdots\cup T_{r-1}$ such that for each $i\in[r-1]$ , $T_{i}\neq\emptyset$ and $A\setminus T_{i}$ is a non-own subset of $A$ , then $\mathcal{F}$ contains an $r$ -focal hypergraph with focus $A$ .
(ii)

If $A,A_{1},\ldots,A_{r-1}\in\mathcal{F}$ form an $r$ -focal hypergraph with focus $A$ , then the $r-1$ members of $\{A\setminus A_{i}:i\in[r-1]\}$ are pairwise disjoint subsets of $A$ .

Since the observation follows fairly straightforwardly from the definition of an $r$ -focal hypergraph, we omit its proof. For later reference, we remark that 2.1 (i) and (ii) will be used in the proofs of the upper and lower bounds of the limit in Theorem 1.2, respectively.

Throughout this section, we will use the following notation. For fixed $r,k$ , let $t:=\lceil\frac{(r-2)k}{r-1}\rceil$ . Then $t=k-\lfloor\frac{k}{r-1}\rfloor$ . Moreover, $\lambda\in[r-1]$ satisfies $k+\lambda\equiv 0\pmod{r-1}$ if and only if

\displaystyle\lambda(k-t)+(r-1-\lambda)(k-t+1)=k.

(6)

2.1 The upper bound of $f_{r}(n,k)$

In this subsection, we will prove the following upper bound.

Theorem 2.2.

For $r\geq 3$ and $k\geq 2$ , let $t=\lceil\frac{(r-2)k}{r-1}\rceil$ . When $n\geq\max\{k,n_{0}\}$ , where $n_{0}=\left(\binom{k}{t}-m(k,k-t,\lambda)\right)t+t-1$ , we have

f_{r}(n,k)\leq\frac{\binom{n}{t}}{\binom{k}{t}-m(k,k-t,\lambda)},

(7)

where $\lambda\in[r-1]$ is the unique integer that satisfies $k+\lambda\equiv 0\pmod{r-1}$ .

The following lemma is needed for the proof of Theorem 2.2.

Lemma 2.3.

Let $n,k$ and $r$ be integers with $n\geq k\geq 2$ and $r\geq 3$ . Let ${\mathcal{F}}\subseteq\binom{[n]}{k}$ be an $r$ -focal-free hypergraph and ${\mathcal{F}}_{0}=\{A\in{\mathcal{F}}:\text{$A$ has no own $(t-1)$-subsets with respect to ${\mathcal{F}}$}\}$ , where $t=\lceil\frac{(r-2)k}{r-1}\rceil$ . Then every $A\in{\mathcal{F}}_{0}$ contains at least $\binom{k}{t}-m(k,k-t,\lambda)$ own $t$ -subsets with respect to ${\mathcal{F}}$ .

Proof.

It suffices to show that every $A\in{\mathcal{F}}_{0}$ contains at most $m(k,k-t,\lambda)$ non-own $t$ -subsets. Suppose on the contrary that there exists some $A\in{\mathcal{F}}_{0}$ that contains at least $m(k,k-t,\lambda)+1$ non-own $t$ -subsets. Define

{\mathcal{F}}_{A}=\{A\setminus B:\ B\text{ is a non-own }t\text{-subset of }A\}\subseteq\binom{A}{k-t}.

By the definition of $m(k,k-t,\lambda)$ , ${\mathcal{F}}_{A}$ contains $\lambda$ pairwise disjoint members, say $T_{1},\ldots,T_{\lambda}\in\binom{A}{k-t}$ . By (6), there exist $r-\lambda-1$ disjoint subsets $T_{\lambda+1},\dots,T_{r-1}\in\binom{A}{k-t+1}$ such that

A=T_{1}\cup T_{2}\cup\cdots\cup T_{r-1}.

According to the definition of ${\mathcal{F}}_{A}$ and the assumption that $A\in{\mathcal{F}}_{0}$ , it is not hard to infer that all of $A\setminus T_{i}$ , $i\in[r-1]$ , are non-own subsets of $A$ . Therefore, it follows from 2.1 (i) that $\mathcal{F}$ contains an $r$ -focal hypergraph with focus $A$ , a contradiction. ∎

Now we are ready to prove Theorem 2.2.

Proof of Theorem 2.2.

Suppose that ${\mathcal{F}}\subseteq\binom{[n]}{k}$ is an $r$ -focal-free hypergraph. Let ${\mathcal{F}}_{0}$ be defined as in Lemma 2.3 and let

{\mathcal{F}}_{1}=\{A\in{\mathcal{F}}:A\text{ contains at least one own $\left(t-1\right)$-subset with respect to }{\mathcal{F}}\}.

Then ${\mathcal{F}}={\mathcal{F}}_{0}\cup{\mathcal{F}}_{1}$ . For each $A\in{\mathcal{F}}_{1}$ , let

\mathcal{O}_{A}:=\left\{T\in\binom{A}{t-1}:T\text{ is an own $\left(t-1\right)$-subset of $A$ with respect to ${\mathcal{F}}$}\right\},

and

\mathcal{B}_{A}:=\left\{B\in\binom{[n]}{t}:\text{$B$ contains some member in $\mathcal{O}_{A}$}.\right\}

Clearly, $A\in{\mathcal{F}}_{1}$ implies that $\mathcal{O}_{A}$ and $\mathcal{B}_{A}$ are both nonempty. Observe that for distinct $A,A^{\prime}\in\mathcal{F}_{1}$ , we have $\mathcal{O}_{A}\cap\mathcal{O}_{A^{\prime}}=\emptyset$ ; hence $|\cup_{A\in{\mathcal{F}}_{1}}\mathcal{O}_{A}|\geq|{\mathcal{F}}_{1}|$ . Note further that any $t$ -set contains $t$ subsets of cardinality $t-1$ , and that each $T\in\mathcal{O}_{A}$ is contained in exactly $n-t+1$ members of $\binom{[n]}{t}$ which belong to $\mathcal{B}_{A}$ . By counting the size of the set

\{(T,B):T\in\bigcup\nolimits_{A\in{\mathcal{F}}_{1}}\mathcal{O}_{A}\text{, }B\in\bigcup\nolimits_{A\in{\mathcal{F}}_{1}}\mathcal{B}_{A}\text{ and }T\subseteq B\}

in two ways, one can infer that

\left|\bigcup\nolimits_{A\in{\mathcal{F}}_{1}}\mathcal{B}_{A}\right|\geq\left|\bigcup\nolimits_{A\in{\mathcal{F}}_{1}}\mathcal{O}_{A}\right|\cdot\frac{n-t+1}{t}\geq|{\mathcal{F}}_{1}|\cdot\frac{n-t+1}{t}.

(8)

For each $M\in{\mathcal{F}}_{0}$ , let $\mathcal{C}_{M}$ be the set consisting of all own $t$ -subsets of $M$ . By Lemma 2.3, we have $|\mathcal{C}_{M}|\geq\binom{k}{t}-m(k,k-t,\lambda)$ . By the definition of $\mathcal{C}_{M}$ , it is easy to see that all of the sets $\mathcal{C}_{M},~{}M\in{\mathcal{F}}_{0}$ , are pairwise disjoint. Therefore,

\left|\bigcup\nolimits_{M\in{\mathcal{F}}_{0}}\mathcal{C}_{M}\right|\geq|{\mathcal{F}}_{0}|\cdot\left(\binom{k}{t}-m(k,k-t,\lambda)\right).

(9)

By definition, $\cup_{A\in{\mathcal{F}}_{1}}\mathcal{B}_{A}$ and $\cup_{M\in{\mathcal{F}}_{0}}\mathcal{C}_{M}$ are also disjoint. Combining (8) and (9) yields that

	$\displaystyle\binom{n}{t}\geq$	$\displaystyle\left\|\bigcup\nolimits_{A\in{\mathcal{F}}_{1}}\mathcal{B}_{A}\right\|+\left\|\bigcup\nolimits_{M\in{\mathcal{F}}_{0}}\mathcal{C}_{M}\right\|$
	$\displaystyle\geq$	$\displaystyle\|{\mathcal{F}}_{1}\|\cdot\frac{n-t+1}{t}+\|{\mathcal{F}}_{0}\|\cdot\left(\binom{k}{t}-m(k,k-t,\lambda)\right)$
	$\displaystyle\geq$	$\displaystyle(\|{\mathcal{F}}_{1}\|+\|{\mathcal{F}}_{0}\|)\cdot\left(\binom{k}{t}-m(k,k-t,\lambda)\right)$
	$\displaystyle=$	$\displaystyle\|{\mathcal{F}}\|\cdot\left(\binom{k}{t}-m(k,k-t,\lambda)\right),$

as needed, where the last inequality holds when $n\geq\left(\binom{k}{t}-m(k,k-t,\lambda)\right)t+t-1.$ ∎

Remark 2.4.

Using a similar method, one can get rid of the assumption $n\geq(\binom{k}{t}-m(k,k-t,\lambda))t+t-1$ , and prove a slightly weaker upper bound with a worse lower order term:

f_{r}(n,k)\leq\frac{\binom{n}{t}}{\binom{k}{t}-m(k,k-t,\lambda)}+\binom{n}{t-1}.

In fact, it is clear that $|\mathcal{F}|=|\mathcal{F}_{0}|+|\mathcal{F}_{1}|$ . Moreover, it is not hard to deduce from the above proof that $|\mathcal{F}_{1}|\leq\binom{n}{t-1}$ and $|\mathcal{F}_{0}|\leq\frac{\binom{n}{t}}{\binom{k}{t}-m(k,k-t,\lambda)}$ , yielding the desired upper bound.

Furthermore, when $r-1\mid k+1$ , i.e., $\lambda=1$ , the upper bound (7) becomes $\binom{n}{\lceil\frac{(r-2)k}{r-1}\rceil}/\binom{k}{\lceil\frac{(r-2)k}{r-1}\rceil}$ . In what follows, we will show that this upper bound can actually be achieved whenever an $(n,k,\lceil\frac{(r-2)k}{r-1}\rceil)$ -design exists.

Proof of Proposition 1.3.

In Introduction, we have already showed that an $(n,k,\lceil\frac{(r-2)k}{r-1}\rceil)$ -packing is also $r$ -focal-free. The above discussion and the upper bound (7) for the special case $\lambda=1$ together prove the first half of the proposition.

For the second half, just note that Keevash [21] proved that $(n,k,t)$ -designs always exist whenever $n$ is large enough and satisfies $\binom{k-i}{t-i}\mid\binom{n-i}{t-i}$ for all $0\leq i\leq t-1$ (see also [8, 18] for alternative proofs). ∎

2.2 The lower bound of $f_{r}(n,k)$

In this subsection, we aim to prove the following lower bound.

Theorem 2.5.

For $r\geq 3$ and $k\geq 2$ , let $t=\lceil\frac{(r-2)k}{r-1}\rceil$ . Then we have

f_{r}(n,k)\geq(1-o(1))\cdot\frac{\binom{n}{t}}{\binom{k}{t}-m(k,k-t,\lambda)},

(10)

where $\lambda\in[r-1]$ is the unique integer that satisfies $k+\lambda\equiv 0\pmod{r-1}$ , and $o(1)\to 0$ as $n\to\infty$ .

To prove Theorem 2.5, below we will introduce packings and induced packings of hypergraphs.

Definition 2.6 (Packings and induced packings).

For a fixed $t$ -uniform hypergraph ${\mathcal{F}}$ and a host $t$ -uniform hypergraph ${\mathcal{H}}$ , a family of $m$ $t$ -uniform hypergraphs

\{(V({\mathcal{F}}_{1}),{\mathcal{F}}_{1}),~{}(V({\mathcal{F}}_{2}),{\mathcal{F}}_{2}),\ldots,(V({\mathcal{F}}_{m}),{\mathcal{F}}_{m})\}

forms an ${\mathcal{F}}$ -packing in ${\mathcal{H}}$ if for each $j\in[m]$ ,

(i)

$V({\mathcal{F}}_{j})\subseteq V({\mathcal{H}})$ , ${\mathcal{F}}_{j}\subseteq{\mathcal{H}}$ ;
(ii)

${\mathcal{F}}_{j}$ is a copy of ${\mathcal{F}}$ defined on the vertex set $V({\mathcal{F}}_{j})$ ;
(iii)

The $m$ $\mathcal{F}$ -copies are pairwise edge disjoint, i.e., ${\mathcal{F}}_{i}\cap{\mathcal{F}}_{j}=\emptyset$ for arbitrary distinct $i,j\in[m]$ .

The above ${\mathcal{F}}$ -packing is said to be induced if it further satisfies

(iv)

$|V({\mathcal{F}}_{i})\cap V({\mathcal{F}}_{j})|\leq t$ for arbitrary distinct $i,j\in[m]$ ;
(v)

For $i\neq j$ , if $|V({\mathcal{F}}_{i})\cap V({\mathcal{F}}_{j})|=t$ , then $V({\mathcal{F}}_{i})\cap V({\mathcal{F}}_{j})\notin{\mathcal{F}}_{i}\cup{\mathcal{F}}_{j}$ .

For our purpose, it suffices to use the above definition with $\mathcal{H}=\binom{[n]}{t}$ . Since the $t$ -uniform hypergraphs in an $\mathcal{F}$ -packing are pairwise edge disjoint, it is clear that every $\mathcal{F}$ -packing in $\binom{[n]}{k}$ can have at most $\binom{n}{t}/|\mathcal{F}|$ copies of $\mathcal{F}$ . The influential works of Rödl [26], Frankl and Rödl [17], and Pippenger (see [24]) showed that the upper bound is asymptotically tight in the sense that there exists a near-optimal $\mathcal{F}$ -packing that contains at least $(1-o(1))\cdot\binom{n}{t}/|\mathcal{F}|$ edge disjoint copies of $\mathcal{F}$ . Frankl and Füredi [15] strengthened their results by showing that there exists a near-optimal induced $\mathcal{F}$ -packing. Their result turns out to be quite useful. It has been used to determine asymptotically the extremal number (or the degenerate Turán density) for several hypergraph extremal problems, see [3, 15, 27] for a few examples.

We quote the result of Frankl and Füredi as follows.

Lemma 2.7 ([15]).

Let $k>t$ and $\mathcal{F}\subseteq\binom{[k]}{t}$ be fixed. Then there exists an induced ${\mathcal{F}}$ -packing $\{(V({\mathcal{F}}_{i}),{\mathcal{F}}_{i}):i\in[m]\}$ in $\binom{[n]}{t}$ with $m\geq(1-o(1))\cdot\binom{n}{t}/|\mathcal{F}|$ , where $o(1)\to 0$ as $n\to\infty$ .

We proceed to present the proof of Theorem 2.5.

Proof of Theorem 2.5.

Let ${\mathcal{G}}\subseteq\binom{[k]}{k-t}$ be one of the largest $(k-t)$ -uniform hypergraphs on $k$ vertices that do not contain $\lambda$ pairwise disjoint edges, where $t=\lceil\frac{(r-2)k}{r-1}\rceil$ . By definition, we have $|{\mathcal{G}}|=m(k,k-t,\lambda)$ . Let ${\mathcal{G}}^{\prime}=\{[k]\setminus A:A\in{\mathcal{G}}\}$ and ${\mathcal{F}}=\binom{[k]}{t}\setminus{\mathcal{G}}^{\prime}$ . Clearly, we have $|{\mathcal{G}}^{\prime}|=|{\mathcal{G}}|$ and $|{\mathcal{F}}|=\binom{k}{t}-|{\mathcal{G}}^{\prime}|=\binom{k}{t}-m(k,k-t,\lambda)$ .

Applying Lemma 2.7 with ${\mathcal{F}}$ defined as above gives an induced ${\mathcal{F}}$ -packing $\{(V({\mathcal{F}}_{i}),{\mathcal{F}}_{i}):i\in[m]\}$ in $\binom{[n]}{t}$ with

m\geq(1-o(1))\cdot\binom{n}{t}/|\mathcal{F}|=(1-o(1))\cdot\frac{\binom{n}{t}}{\binom{k}{t}-m(k,k-t,\lambda)},

where $o(1)\to 0$ as $n\to\infty$ . Note that for each $i\in[m]$ , we have $V({\mathcal{F}}_{i})\in\binom{[n]}{k}$ .

The following observation follows straightforwardly from the construction of ${\mathcal{F}}$ .

Observation 2.8.

We have ${\mathcal{G}}=\{[k]\setminus A^{\prime}:A^{\prime}\in\binom{[k]}{t}\setminus{\mathcal{F}}\}$ . By definition of ${\mathcal{G}}$ , for every copy ${\mathcal{F}}_{i}$ of ${\mathcal{F}}$ , the $(k-t)$ -uniform hypergraph $\{V({\mathcal{F}}_{i})\setminus A^{\prime}:A^{\prime}\in\binom{V({\mathcal{F}}_{i})}{t}\setminus{\mathcal{F}}_{i}\}$ does not contain $\lambda$ pairwise disjoint edges.

To prove Theorem 2.5, it suffices to show that

{\mathcal{H}}({\mathcal{F}}):=\{V({\mathcal{F}}_{i}):i\in[m]\}\subseteq\binom{[n]}{k}

is $r$ -focal-free. Suppose for the sake of contradiction that $V({\mathcal{F}}_{1}),\ldots,V({\mathcal{F}}_{r})\in{\mathcal{H}}({\mathcal{F}})$ form an $r$ -focal hypergraph with focus $V({\mathcal{F}}_{r})$ . Then, it follows from 2.1 (ii) that all members of $\{V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{i}):i\in[r-1]\}$ are pairwise disjoint subsets of $V({\mathcal{F}}_{r})$ . Moreover, it follows from the definition of an induced packing (see Definition 2.6 (iv)) that for each $i\in[r-1]$ , $|V({\mathcal{F}}_{r})\cap V({\mathcal{F}}_{i})|\leq t$ , which implies that $|V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{i})|\geq k-t$ . Combining the discussion above and (6), it is not hard to verify that there are at least $\lambda$ distinct $i\in[r-1]$ such that $|V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{i})|=k-t$ . Assume without loss of generality that

|V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{1})|=\cdots=|V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{\lambda})|=k-t.

This implies that for each $i\in[\lambda]$ , we have $|V({\mathcal{F}}_{r})\cap V({\mathcal{F}}_{i})|=t$ .

It again follows from the definition of an induced packing (see Definition 2.6 (v)) that for each $i\in[\lambda]$ , $V({\mathcal{F}}_{r})\cap V({\mathcal{F}}_{i})\not\in{\mathcal{F}}_{r}$ , and hence $\{V({\mathcal{F}}_{r})\cap V({\mathcal{F}}_{i}):i\in[\lambda]\}\subseteq\binom{V({\mathcal{F}}_{r})}{t}\setminus{\mathcal{F}}_{r}$ . Consequently, we have

\{V({\mathcal{F}}_{r})\setminus V({\mathcal{F}}_{i}):i\in[\lambda]\}\subseteq\{V({\mathcal{F}}_{r})\setminus A^{\prime}:A^{\prime}\in\binom{V({\mathcal{F}}_{r})}{t}\setminus{\mathcal{F}}_{r}\}

and therefore the latter hypergraph contains $\lambda$ pairwise disjoint edges, which contradicts 2.8. This completes the proof of Theorem 2.5. ∎

3 Focal-free codes

3.1 Upper and lower bounds of $f_{r}^{q}(n)$

The proof of Theorem 1.5 follows from the same approach as the proof of Theorem 1.2, which is briefly explained below.

Let us define the own subsequences of a vector, which is an analogy for own subsets of a set. For a vector $\bm{x}=(x_{1},\ldots,x_{n})\in[q]^{n}$ and a subset of indices $T\subseteq[n]$ , let $\bm{x}_{T}=(x_{i}:i\in T)$ denote the subsequence of $\bm{x}$ with coordinates indexed by $T$ . For two vectors $\bm{x},\bm{y}\in[q]^{n}$ , let $I(\bm{x},\bm{y})=\{i\in[n]:x_{i}=y_{i}\}$ denote the set of indices of coordinates for which $\bm{x}$ and $\bm{y}$ are equal. For a code $\mathcal{C}\subseteq[q]^{n}$ and a codeword $\bm{x}\in\mathcal{C}$ , a subsequence $\bm{x}_{T}$ is called an own subsequence of $\bm{x}$ (with respect to ${\mathcal{C}}$ ) if for every $\bm{y}\in{\mathcal{C}}\setminus\{\bm{x}\}$ , $\bm{x}_{T}\neq\bm{y}_{T}$ ; otherwise, $\bm{x}_{T}$ is called a non-own subsequence of $\bm{x}$ (with respect to ${\mathcal{C}}$ ).

Similarly to 2.1, the following observation presents sufficient and necessary conditions for the existence of an $r$ -focal code.

Observation 3.1.

Let $\mathcal{C}\subseteq[q]^{n}$ be a code with $|\mathcal{C}|\geq r$ . Then the following hold:

(i)

If for some $\bm{x}\in\mathcal{C}$ , there is a partition $[n]=T_{1}\cup\cdots\cup T_{r-1}$ such that for each $i\in[r-1]$ , $T_{i}\neq\emptyset$ and $\bm{x}_{[n]\setminus T_{i}}$ is a non-own subsequence of $\bm{x}$ , then $\mathcal{C}$ contains an $r$ -focal code with focus $\bm{x}$ .
(ii)

If $\bm{x},\bm{x}^{1},\dots,\bm{x}^{r-1}\in\mathcal{C}$ form an $r$ -focal code with focus $\bm{x}$ , then the $r-1$ members of $\{[n]\setminus I(\bm{x},\bm{x}^{i}):i\in[r-1]\}$ are pairwise disjoint subsets of $[n]$ .

Since the observation follows directly from the definition of an $r$ -focal code, we omit its proof.

We note that similarly to the usage of 2.1 in the proof of Theorem 1.2, 3.1 (i) and (ii) will also be used in the proofs of the upper and lower bounds of the limit in Theorem 1.5, respectively. The main technical difference is that, instead of dealing with own/non-own subsets, we will work with own/non-own subsequences, and will use an appropriate version of Lemma 2.7 recently developed in [22]. We will state our main results below, and postpone their proofs to Appendix A.

First, the following is an upper bound on $f_{r}^{q}(n)$ .

Theorem 3.2.

For integers $r\geq 3$ and $n\geq 2$ , let $t=\lceil\frac{(r-2)n}{r-1}\rceil$ . When $q\geq\frac{t}{n-t+1}\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$ , we have

f_{r}^{q}(n)\leq\frac{\binom{n}{t}}{\binom{n}{t}-m(n,n-t,\lambda)}q^{t},

where $\lambda\in[r-1]$ is the unique integer that satisfies $n+\lambda\equiv 0\pmod{r-1}$ .

Remark 3.3.

Similarly to Remark 2.4, one can get rid of the assumption $q\geq\frac{t}{n-t+1}(\binom{n}{t}-m(n,n-t,\lambda))$ , and prove a slightly weaker upper bound with a worse lower order term:

f_{r}^{q}(n)\leq\frac{\binom{n}{t}}{\binom{n}{t}-m(n,n-t,\lambda)}q^{t}+\binom{n}{t-1}q^{t-1}.

We omit the details.

We proceed to state a lower bound on $f_{r}^{q}(n)$ .

Theorem 3.4.

For any integers $r\geq 3$ and $n\geq 2$ , let $t=\lceil\frac{(r-2)n}{r-1}\rceil$ . Then we have

f_{r}^{q}(n)\geq(1-o(1))\cdot\frac{\binom{n}{t}}{\binom{n}{t}-m(n,n-t,\lambda)}q^{t},

where $\lambda\in[r-1]$ is the unique integer that satisfies $n+\lambda\equiv 0\pmod{r-1}$ , and $o(1)\to 0$ as $q\to\infty$ .

3.2 Exact values of $f_{r}^{q}(n)$

In this subsection, we will prove Theorem 1.6. First, note that $m(n,\lfloor\frac{n}{r-1}\rfloor,1)=0$ for $\lambda=1$ . By Theorem 3.2, for $n\equiv-1\pmod{r-1}$ and sufficiently large $q\geq q_{0}(n)$ , where $q_{0}(n)=2^{\Theta(n)}$ , we have $f_{r}^{q}(n)\leq q^{\lceil\frac{(r-2)n}{r-1}\rceil}$ . The following theorem, which is the main technical result of this subsection, shows that the same upper bound in fact holds for significantly smaller $q$ .

Theorem 3.5.

Let $r\geq 3$ . Suppose that $q\geq r-1$ and $r-1\mid n+1$ . Then

\displaystyle f_{r}^{q}(n)\leq q^{\lceil\frac{(r-2)n}{r-1}\rceil}.

Clearly, Theorem 3.5 implies the upper bound of Theorem 1.6. We postpone the proof of Theorem 3.5 to the end of this subsection.

Now we turn to the lower bound of Theorem 1.6. Indeed, Alon and Holzman [1] proved exactly the same lower bound under a stronger assumption on the parameters, i.e., $q\geq n$ and $q$ is a prime power (see (4)). We will prove our new result by connecting error-correcting codes with large minimum distance to focal-free codes. Note that such a connection was implicitly used in the proof of (4) in [1] (see Proposition 3.2 in [1]). We will state it explicitly in the next lemma. Note that for any two vectors $\bm{x},\bm{y}\in[q]^{n}$ , the Hamming distance between $\bm{x},\bm{y}$ is defined by $d(\bm{x},\bm{y})=|\{i\in[n]:x_{i}\neq y_{i}\}|$ . The minimum distance of a code ${\mathcal{C}}\subseteq[q]^{n}$ , denoted by $d({\mathcal{C}})$ , is $\min\{d(\bm{x},\bm{y}):\text{~{}distinct~{}}\bm{x},\bm{y}\in{\mathcal{C}}\}$ .

Lemma 3.6.

If ${\mathcal{C}}\subseteq[q]^{n}$ satisfies $d({\mathcal{C}})>\lfloor\frac{n}{r-1}\rfloor$ , then ${\mathcal{C}}$ is $r$ -focal-free.

Proof.

Assume otherwise that $\{\bm{x}^{0},\bm{x}^{1},\dots,\bm{x}^{r-1}\}\subseteq{\mathcal{C}}$ form an $r$ -focal code with focus $\bm{x}^{0}$ . Then by 3.1 (ii), $\{[n]\setminus I(\bm{x}^{0},\bm{x}^{i}):i\in[r-1]\}$ are pairwise disjoint subsets of $[n]$ , which implies that $\sum_{j=1}^{r-1}d(\bm{x}^{0},\bm{x}^{j})\leq n$ . Therefore, there exists some $j\in[r-1]$ such that $d(\bm{x}^{0},\bm{x}^{j})\leq\lfloor\frac{n}{r-1}\rfloor$ , a contradiction. ∎

Alon and Holzman constructed focal-free codes by Reed-Solomon codes [25], which is a classic in coding theory. Here, we will construct focal-free codes by applying Lemma 3.6 to codes generated by orthogonal arrays. More precisely, given positive integers $t,n,q$ , an orthogonal array $OA(t,n,q)$ is an $n\times q^{t}$ matrix, say $A$ , with entries from $[q]$ such that in every $t\times q^{t}$ submatrix of $A$ , every possible vector in $[q]^{t}$ appears as a column exactly once. Hence, any two different columns of $A$ agree in at most $t-1$ rows. Let $\mathcal{C}\subseteq[q]^{n}$ be the code formed by the column vectors of $A$ . Then, it is easy to see that $d(\mathcal{C})\geq n-t+1$ . Consequently, we can construct the codes required in Lemma 3.6 by known results on the existence of orthogonal arrays.

Lemma 3.7 (see [7, Theorems III.7.18 and III.7.20, page 226]).

Let $q=p_{1}^{e_{1}}\cdots p_{s}^{e_{s}}$ be the canonical integer factorization of $q\geq 2$ , where $p_{1},\ldots,p_{s}$ are distinct primes, $e_{1},\ldots,e_{s}$ are positive integers, and $p_{1}^{e_{1}}<\cdots<p_{s}^{e_{s}}$ . If $t<p_{1}^{e_{1}}$ and $n\leq p_{1}^{e_{1}}+1$ , then an $OA(t,n,q)$ exists.

We prove Theorem 1.6 by combining Theorem 3.5 and Lemmas 3.6 and 3.7.

Proof of Theorem 1.6.

The upper bound $f_{r}^{q}(n)\leq q^{\lceil\frac{(r-2)n}{r-1}\rceil}$ is a direct consequence of Theorem 3.5. For the lower bound, according to above discussion and Lemma 3.6, it is not hard to see that $f_{r}^{q}(n)\geq q^{\lceil\frac{(r-2)n}{r-1}\rceil}$ as long as there exists an $OA(\lceil\frac{(r-2)n}{r-1}\rceil,n,q)$ . In the meantime, the existence of such an orthogonal array follows straightforwardly from Lemma 3.7. ∎

It remains to prove Theorem 3.5.

Proof of Theorem 3.5.

Write $n=(r-1)d-1$ . Then $\lceil\frac{(r-2)n}{r-1}\rceil=n-d+1$ . It suffices to show that $f_{r}^{q}(n)\leq q^{n-d+1}$ . Suppose for the sake of contradiction that there exists an $r$ -focal-free code ${\mathcal{C}}\subseteq[q]^{n}$ with $|{\mathcal{C}}|\geq q^{n-d+1}+1$ . Clearly, $d\geq 2$ since $|{\mathcal{C}}|\leq q^{n}$ . For a subset $S\subseteq[n]$ , let

U_{S}=\{\bm{x}\in\mathcal{C}:\text{$\bm{x}_{S}$ is an own subsequence of $\bm{x}$ with respect to $\mathcal{C}$}\}.

Then, 3.1 (i) implies that for every partition $[n]=T_{1}\cup\cdots\cup T_{r-1}$ , where $T_{i}\neq\emptyset$ for each $i\in[r-1]$ , we have

\displaystyle\mathcal{C}=U_{[n]\setminus T_{1}}\cup\cdots\cup U_{[n]\setminus T_{r-1}}.

(11)

Let $S\in\binom{[n]}{n-d}$ be a subset such that $|U_{S}|$ is maximized (possibly $|U_{S}|=0$ , and break ties arbitrarily). We will deduce a contradiction by showing that if $q\geq r-1$ and $|{\mathcal{C}}|\geq q^{n-d+1}+1$ , then there must exist some $S^{\prime}\in\binom{[n]}{n-d}\setminus\{S\}$ such that $|U_{S^{\prime}}|\geq|U_{S}|+1$ .

Assume without loss of generality that $S=[n-d]$ . We will construct the desired $S^{\prime}$ by bounding $|U_{[n-d+1]}|$ from the above. Clearly,

U_{[n-d+1]}=(U_{[n-d+1]}\cap U_{[n-d]})\cup(U_{[n-d+1]}\setminus U_{[n-d]}).

It is obvious that $|U_{[n-d+1]}\cap U_{[n-d]}|\leq|U_{[n-d]}|$ . Moreover, for every $\bm{y}\in[q]^{n-d}$ (which is not an own subsequence $\bm{x}_{[n-d]}$ for any $\bm{x}\in{\mathcal{C}}$ ), there are at most $q$ choices of $\bm{x}\in{\mathcal{C}}$ such that there exists some $a\in[q]$ , for which $(\bm{y},a)\in[q]^{n-d+1}$ is an own subsequence $\bm{x}_{[n-d+1]}$ of $\bm{x}$ . This implies that

|U_{[n-d+1]}\setminus U_{[n-d]}|\leq q\cdot(q^{n-d}-|U_{[n-d]}|).

Therefore, we have

\displaystyle|U_{[n-d+1]}|\leq|U_{[n-d]}|+q\cdot(q^{n-d}-|U_{[n-d]}|)=q^{n-d+1}-(q-1)\cdot|U_{[n-d]}|.

(12)

Observe that $n=(r-2)d+(d-1)$ . Consider the partition $[n]=T_{1}\cup\dots\cup T_{r-1}$ , where $T_{i}=\{(i-1)d+1,\dots,id\}$ for $i\in[r-2]$ and $T_{r-1}=\{(r-2)d+1,\dots,(r-1)d-1\}$ . Then $|T_{1}|=\dots=|T_{r-2}|=d,~{}|T_{r-1}|=d-1$ , and $[n-d+1]=[n]\setminus T_{r-1}$ . By (11) and (12), we have

	$\displaystyle\|U_{[n]\setminus T_{1}}\cup\cdots\cup U_{[n]\setminus T_{r-2}}\|$	$\displaystyle\geq\|{\mathcal{C}}\|-\|U_{[n]\setminus T_{r-1}}\|=\|{\mathcal{C}}\|-\|U_{[n-d+1]}\|$
		$\displaystyle\geq(\|{\mathcal{C}}\|-q^{n-d+1})+(q-1)\cdot\|U_{[n-d]}\|$
		$\displaystyle\geq 1+(q-1)\cdot\|U_{[n-d]}\|.$

By pigeonhole principle, there exists some $i\in[r-2]$ such that

|U_{[n]\setminus T_{i}}|\geq\left\lceil\frac{1+(q-1)\cdot|U_{[n-d]}|}{r-2}\right\rceil\geq|U_{[n-d]}|+1,

where the last inequality holds since $q-1\geq r-2$ . Setting $S^{\prime}=[n]\setminus T_{i}$ , we have $S^{\prime}\in\binom{[n]}{n-d}$ and $|U_{S^{\prime}}|>|U_{[n-d]}|$ , contradicting the maximality of $U_{[n-d]}$ . This completes the proof of the theorem. ∎

4 Concluding remarks

In this paper, we presented asymptotically tight upper and lower bounds for both focal-free uniform hypergraphs and codes, thus improving the corresponding results of Alon and Holzman [1]. In addition, we also determined the exact values of these hypergraphs and codes for infinitely many parameters. Many interesting problems remain open.

•

Let $f_{r}(n)$ denote the maximum cardinality of an $r$ -focal-free hypergraph ${\mathcal{F}}\subseteq 2^{[n]}$ . In [1], it was observed that the upper and lower bounds on $f_{r}(n)$ can be proved by $f_{r}(n)\leq\sum_{k=1}^{n}f_{r}(n,k)$ and by the probabilistic method, respectively. Can we improve these bounds?
•

We have determined $f_{r}^{q}(n)$ asymptotically for fixed $r,n$ and $q\rightarrow\infty$ , However, for fixed $r,q$ and $n\rightarrow\infty$ , there is still a huge gap between the known upper and lower bounds (see [1] for details). So, it would be interesting to narrow this gap. In particular, the upper bound for the binary case is also an upper bound for near-sunflower-free hypergraphs.
•

Theorems 2.5 and 3.4 are proved by Lemmas 2.7 and A.3, and therefore the lower bounds are non-explicit. Giving near-optimal explicit constructions for both problems remains open.

Acknowledgements

The authors would like to thank Zixiang Xu for telling them that this work was initiated by Xinqi Huang, Xiande Zhang and Yuhao Zhao, and by Chong Shangguan simultaneously and independently, which led to this collaboration. The research of Xiande Zhang is supported by the National Key Research and Development Programs of China 2023YFA1010200 and 2020YFA0713100, the NSFC under Grants No. 12171452 and No. 12231014, and the Innovation Program for Quantum Science and Technology 2021ZD0302902. The research of Chong Shangguan is supported by the National Natural Science Foundation of China under Grant Nos. 12101364 and 12231014, and the Natural Science Foundation of Shandong Province under Grant No. ZR2021QA005.

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Appendix A Proof of Theorem 1.5

In this section, we present proofs of Theorems 3.2 and 3.4. For fixed $r,n$ , let $t:=\lceil\frac{(r-2)n}{r-1}\rceil$ . Then $t=n-\lfloor\frac{n}{r-1}\rfloor$ . Moreover, $\lambda\in[r-1]$ satisfies $n+\lambda\equiv 0\pmod{r-1}$ if and only if

\displaystyle\lambda(n-t)+(r-1-\lambda)(n-t+1)=n.

(13)

For notational convenience, we need the following useful definition that connects subsets of $[q]^{n}$ to multi-partite hypergraphs.

Definition A.1.

For positive integers $q,n$ , let ${\mathcal{H}}_{n}(q)$ denote the complete $n$ -partite $n$ -uniform hypergraph with equal part size $q$ , where the vertex set $V({\mathcal{H}}_{n}(q))$ admits a partition $V({\mathcal{H}}_{n}(q))=V_{1}\cup\cdots\cup V_{n}$ with $V_{i}=\{(i,a):a\in[q]\}$ for each $i\in[n]$ , and the edge set is defined as

{\mathcal{H}}_{n}(q)=\{\{(1,a_{1}),\ldots,(n,a_{n})\}:a_{1},a_{2},\ldots,a_{n}\in[q]\}.

Clearly, $|V({\mathcal{H}}_{n}(q))|=nq$ and $|{\mathcal{H}}_{n}(q)|=q^{n}$ .

For $t\leq n$ , let ${\mathcal{H}}_{n}^{(t)}(q)$ denote the complete $n$ -partite $t$ -uniform hypergraph with equal part size $q$ , where $V({\mathcal{H}}_{n}^{(t)}(q))=V({\mathcal{H}}_{n}(q))$ , and

{\mathcal{H}}_{n}^{(t)}(q)=\{\{(i_{1},a_{i_{1}}),\ldots,(i_{t},a_{i_{t}})\}:1\leq i_{1}<\cdots<i_{t}\leq n,a_{i_{1}},\ldots,a_{i_{t}}\in[q]\}.

Then $|{\mathcal{H}}_{n}^{(t)}(q)|=\binom{n}{t}q^{t}$ .

Let $\pi:[q]^{n}\rightarrow\mathcal{H}_{n}(q)$ be a bijection defined as follows. For each $\bm{x}\in[q]^{n}$ , let

\pi(\bm{x}):=\{(1,x_{1}),(2,x_{2}),\dots,(n,x_{n})\}\in\mathcal{H}_{n}(q).

For $\mathcal{C}\subseteq[q]^{n}$ , let $\pi(C):=\{\pi(\bm{x}):\bm{x}\in\mathcal{C}\}$ . Moreover, for $T\subseteq[n]$ and a subsequence $\bm{x}_{T}$ of $\bm{x}$ , let $\pi(\bm{x}_{T})=\{(i,x_{i}):i\in T\}\in\mathcal{H}_{n}^{(|T|)}(q)$ . Crucially, $\pi$ inherits the own subsequence property in the sense that $\bm{x}_{T}$ is an own subsequence of $\bm{x}$ with respect to $\mathcal{C}$ if and only if $\pi(\bm{x}_{T})$ is an own subset of $\pi(\bm{x})$ with respect to $\pi(\mathcal{C})$ .

A.1 Proof of Theorem 3.2

The following lemma is an analogue to Lemma 2.3.

Lemma A.2.

Let $n\geq 2$ and $r\geq 3$ . Suppose that ${\mathcal{C}}\subseteq[q]^{n}$ is an $r$ -focal-free code. Let ${\mathcal{C}}_{0}$ be the set of codewords in ${\mathcal{C}}$ that have no own $(t-1)$ -subsequence with respect to ${\mathcal{C}}$ , where $t=\lceil\frac{(r-2)n}{r-1}\rceil$ . Then every $\bm{x}\in{\mathcal{C}}_{0}$ contains at least $\binom{n}{t}-m(n,n-t,\lambda)$ own $t$ -subsequences with respect to ${\mathcal{C}}$ .

Proof.

It suffices to show that every $\bm{x}\in{\mathcal{C}}_{0}$ contains at most $m(n,n-t,\lambda)$ non-own $t$ -subsequences. Suppose on the contrary that there exists some $\bm{x}\in{\mathcal{C}}_{0}$ that contains at least $m(n,n-t,\lambda)+1$ non-own $t$ -subsequences with respect to ${\mathcal{C}}$ . Let

{\mathcal{F}}_{\bm{x}}:=\{T\in\binom{[n]}{n-t}:\bm{x}_{[n]\setminus T}\text{ is a non-own $t$-subsequence of }\bm{x}\}.

Then $|{\mathcal{F}}_{\bm{x}}|\geq m(n,n-t,\lambda)+1$ and by definition, ${\mathcal{F}}_{\bm{x}}$ contains $\lambda$ pairwise disjoint members $T_{1},\dots,T_{\lambda}$ . By (13), there exist $T_{\lambda+1},\dots,T_{r-1}\in\binom{[n]}{n-t+1}$ such that

[n]=T_{1}\cup\cdots\cup T_{r-1}.

As $T_{1},\dots,T_{\lambda}\in{\mathcal{F}}_{\bm{x}}$ and $\bm{x}\in{\mathcal{C}}_{0}$ , for each $i\in[r-1]$ , $\bm{x}_{[n]\setminus T_{i}}$ is a non-own subsequence of $\bm{x}$ . Therefore, it follows from 3.1 (i) that $\mathcal{C}$ contains an $r$ -focal code with focus $\bm{x}$ , a contradiction. ∎

Now we are ready to present the proof of Theorem 3.2.

Proof of Theorem 3.2.

Suppose that ${\mathcal{C}}\subseteq[q]^{n}$ is an $r$ -focal-free code. Let ${\mathcal{C}}_{0}$ be defined as in Lemma A.2, and let

{\mathcal{C}}_{1}=\{\bm{x}\in{\mathcal{C}}:\bm{x}\text{ contains at least one own $\left(t-1\right)$-subsequence with respect to }{\mathcal{C}}\}.

Clearly, ${\mathcal{C}}={\mathcal{C}}_{0}\cup{\mathcal{C}}_{1}$ . By the discussion above, for each $\bm{x}\in{\mathcal{C}}_{1}$ , $\pi(\bm{x})$ contains at least one own $(t-1)$ -subset with respect to $\pi({\mathcal{C}})$ . Let

\mathcal{O}_{\bm{x}}:=\{T\in{\mathcal{H}}_{n}^{(t-1)}(q):T\text{ is an own $\left(t-1\right)$-subset of $\pi(\bm{x})$ with respect to $\pi({\mathcal{C}})$}\},

and

{\mathcal{B}}_{\bm{x}}:=\{B\in{\mathcal{H}}_{n}^{(t)}(q):B\text{ contains some $T\in\mathcal{O}_{\bm{x}}$}\}.

Clearly, $\bm{x}\in{\mathcal{C}}_{1}$ implies that $\mathcal{O}_{\bm{x}}$ and ${\mathcal{B}}_{\bm{x}}$ are both nonempty. For distinct $\bm{x},\bm{x}^{\prime}\in\mathcal{C}_{1}$ , we have $\mathcal{O}_{\bm{x}}\cap\mathcal{O}_{\bm{x}^{\prime}}=\emptyset$ , so $|\cup_{\bm{x}\in{\mathcal{C}}_{1}}\mathcal{O}_{\bm{x}}|\geq|{\mathcal{C}}_{1}|$ . Moreover, every $T=\{(i_{1},a_{i_{1}}),\ldots,(i_{t-1},a_{i_{t-1}})\}\in\mathcal{O}_{\bm{x}}$ is contained in exactly $(n-t+1)q$ edges in ${\mathcal{H}}_{n}^{(t)}(q)$ , say $\{(i_{1},a_{i_{1}}),\ldots,(i_{t-1},a_{i_{t-1}}),(i_{t},a_{i_{t}})\}\in{\mathcal{H}}_{n}^{(t)}(q)$ , where $i_{t}\in[n]\setminus\{i_{1},\ldots,i_{t-1}\}$ has $n-t+1$ choices and $a_{i_{t}}\in[q]$ has $q$ choices. Therefore, by counting the size of the set

\{(T,B):T\in\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}\mathcal{O}_{\bm{x}}\text{, }B\in\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}\mathcal{B}_{\bm{x}}\text{ and }T\subseteq B\}

in two ways, one can infer that

\left|\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}{\mathcal{B}}_{\bm{x}}\right|\geq\frac{1}{t}\cdot\left|\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}\mathcal{O}_{\bm{x}}\right|(n-t+1)q\geq|{\mathcal{C}}_{1}|\cdot\frac{(n-t+1)q}{t}.

For each $\bm{y}\in{\mathcal{C}}_{0}$ , let

\mathcal{A}_{\bm{y}}:=\{S\in{\mathcal{H}}_{n}^{(t)}(q):S\text{ is an own $t$-subset of $\pi(\bm{y})$ with respect to $\pi({\mathcal{C}})$}\}.

By Lemma A.2, $|\mathcal{A}_{\bm{y}}|\geq\binom{n}{t}-m(n,n-t,\lambda)$ for every $\bm{y}\in{\mathcal{C}}_{0}$ . Note that by the definition of own subsequences and subsets, it is routine to check that $\mathcal{A}_{\bm{y}}$ , $\bm{y}\in\mathcal{C}_{0}$ , are pairwise disjoint; moreover, $(\cup_{\bm{x}\in{\mathcal{C}}_{1}}{\mathcal{B}}_{\bm{x}})\cap(\cup_{\bm{y}\in{\mathcal{C}}_{0}}{\mathcal{A}}_{\bm{y}})=\emptyset$ . Consequently,

	$\displaystyle\binom{n}{t}q^{t}$	$\displaystyle=\|{\mathcal{H}}_{n}^{(t)}(q)\|\geq\left\|\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}{\mathcal{B}}_{\bm{x}}\right\|+\left\|\bigcup\nolimits_{\bm{y}\in{\mathcal{C}}_{0}}{\mathcal{A}}_{\bm{y}}\right\|$
		$\displaystyle\geq\|{\mathcal{C}}_{1}\|\cdot\frac{(n-t+1)q}{t}+\|{\mathcal{C}}_{0}\|\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$
		$\displaystyle\geq(\|{\mathcal{C}}_{1}\|+\|{\mathcal{C}}_{0}\|)\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$
		$\displaystyle=\|{\mathcal{C}}\|\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right),$

where the last inequality holds whenever $q\geq\frac{t}{n-t+1}\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$ , as needed. ∎

A.2 Proof of Theorem 3.4

The main ingredient for the proof of Theorem 3.4 is a version of Lemma 2.7 stated for multi-partite hypergraphs. We need some more definitions before formally stating it.

Suppose ${\mathcal{F}}$ and ${\mathcal{H}}$ are $k$ -partite hypergraphs with vertex partitions $V({\mathcal{F}})=\cup_{i=1}^{k}W_{i}$ and $V({\mathcal{H}})=\cup_{i=1}^{k}V_{i}$ , respectively. A copy ${\mathcal{F}}^{\prime}$ of ${\mathcal{F}}$ in ${\mathcal{H}}$ is called faithful (with respect to the partitions above) if for each $i\in[k]$ , the copy of $W_{i}$ in $V({\mathcal{F}}^{\prime})$ is contained in $V_{i}$ . An ${\mathcal{F}}$ -packing $\{(V({\mathcal{F}}_{i}),{\mathcal{F}}_{i}):i\in[m]\}\subseteq\mathcal{H}$ is said to be faithful, if for every $j\in[m]$ , ${\mathcal{F}}_{j}$ is a faithful copy of ${\mathcal{F}}$ .

Lemma A.3 ([22]).

Let $n>t$ and $\mathcal{F}\subseteq\binom{[n]}{t}$ be fixed. Then viewing $\mathcal{F}$ as an $n$ -partite hypergraph, there exists a faithful induced ${\mathcal{F}}$ -packing $\{(V({\mathcal{F}}_{i}),{\mathcal{F}}_{i}):i\in[m]\}$ in ${\mathcal{H}}_{n}^{(t)}(q)$ with $m\geq(1-o(1))\cdot\frac{\binom{n}{t}q^{t}}{|{\mathcal{F}}|}$ , where $o(1)\rightarrow 0$ as $q\rightarrow\infty$ .

Proof of Theorem 3.4.

Let ${\mathcal{G}}\subseteq\binom{[n]}{n-t}$ be one of the largest $(n-t)$ -uniform hypergraphs on $n$ vertices that do not contain $\lambda$ pairwise disjoint edges, where $t=\lceil\frac{(r-2)n}{r-1}\rceil$ . Then by definition we have $|{\mathcal{G}}|=m(n,n-t,\lambda)$ . Let ${\mathcal{G}}^{\prime}=\{A\subseteq[n]:[n]\backslash A\in{\mathcal{G}}\}$ and ${\mathcal{F}}=\binom{[n]}{t}\backslash{\mathcal{G}}^{\prime}$ . Clearly, $|{\mathcal{G}}|=|{\mathcal{G}}^{\prime}|$ and $|{\mathcal{F}}|=\binom{n}{t}-|{\mathcal{G}}|=\binom{n}{t}-m(n,n-t,\lambda)$ .

Applying Lemma A.3 with ${\mathcal{F}}$ defined as above gives a faithful induced ${\mathcal{F}}$ -packing $\{(V({\mathcal{F}}_{i}),{\mathcal{F}}_{i}):i\in[m]\}$ in ${\mathcal{H}}_{n}^{(t)}(q)$ with

\displaystyle m\geq(1-o(1))\cdot\frac{\binom{n}{t}q^{t}}{|{\mathcal{F}}|}=(1-o(1))\cdot\frac{\binom{n}{t}q^{t}}{\binom{n}{t}-m(n,n-t,\lambda)}.

where $o(1)\to 0$ as $q\to\infty$ . Note that for each $i\in[m]$ , ${\mathcal{F}}_{i}$ is a faithful copy of ${\mathcal{F}}$ , where both ${\mathcal{F}}$ and ${\mathcal{H}}_{n}^{(t)}(q)$ are viewed as $n$ -partite hypergraphs. We can treat each copy $V({\mathcal{F}}_{i})$ of $V({\mathcal{F}})$ as a vector in $[q]^{n}$ , according to (the inverse of) $\pi$ defined below Definition A.1. Let ${\mathcal{C}}=\{\pi^{-1}(V({\mathcal{F}}_{i})):i\in[m]\}$ .

To prove the theorem, it remains to show that ${\mathcal{C}}$ is $r$ -focal-free. Assume for the sake of contradiction that $\{\bm{x}^{1},\ldots,\bm{x}^{r}\}\subseteq{\mathcal{C}}$ forms an $r$ -focal code with focus $\bm{x}^{r}$ . It follows from 3.1 that $[n]\setminus I(\bm{x}^{r},\bm{x}^{1}),\ldots,[n]\setminus I(\bm{x}^{r},\bm{x}^{r-1})$ are $r-1$ pairwise disjoint subsets. Moreover, assume without loss of generality that for each $i\in[r]$ , $\bm{x}^{i}=\pi^{-1}(V(\mathcal{F}_{i}))$ , where $\mathcal{F}_{i}$ is a copy of $\mathcal{F}$ in the previous $\mathcal{F}$ -packing. Then, for each $i\in[r-1]$ , we have $|I(\bm{x}^{r},\bm{x}^{i})|=|V(\mathcal{F}_{r})\cap V(\mathcal{F}_{i})|\leq t$ , which implies that $|[n]\setminus I(\bm{x}^{r},\bm{x}^{i})|\geq n-t$ . Combining the above discussion with (13), one can infer that there are at least $\lambda$ distinct $i\in[r-1]$ such that $|[n]\setminus I(\bm{x}^{r},\bm{x}^{i})|=n-t$ , which implies that $|I(\bm{x}^{r},\bm{x}^{i})|=|V(\mathcal{F}_{r})\cap V(\mathcal{F}_{i})|=t$ . Then we obtain a contradiction due to the same reason as in the proof of Theorem 2.5, we omit the details. ∎

	$\displaystyle\|U_{[n]\setminus T_{1}}\cup\cdots\cup U_{[n]\setminus T_{r-2}}\|$	$\displaystyle\geq\|{\mathcal{C}}\|-\|U_{[n]\setminus T_{r-1}}\|=\|{\mathcal{C}}\|-\|U_{[n-d+1]}\|$
		$\displaystyle\geq(\|{\mathcal{C}}\|-q^{n-d+1})+(q-1)\cdot\|U_{[n-d]}\|$
		$\displaystyle\geq 1+(q-1)\cdot\|U_{[n-d]}\|.$

	$\displaystyle\binom{n}{t}q^{t}$	$\displaystyle=\|{\mathcal{H}}_{n}^{(t)}(q)\|\geq\left\|\bigcup\nolimits_{\bm{x}\in{\mathcal{C}}_{1}}{\mathcal{B}}_{\bm{x}}\right\|+\left\|\bigcup\nolimits_{\bm{y}\in{\mathcal{C}}_{0}}{\mathcal{A}}_{\bm{y}}\right\|$
		$\displaystyle\geq\|{\mathcal{C}}_{1}\|\cdot\frac{(n-t+1)q}{t}+\|{\mathcal{C}}_{0}\|\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$
		$\displaystyle\geq(\|{\mathcal{C}}_{1}\|+\|{\mathcal{C}}_{0}\|)\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right)$
		$\displaystyle=\|{\mathcal{C}}\|\cdot\left(\binom{n}{t}-m(n,n-t,\lambda)\right),$

Focal-free uniform hypergraphs and codes

Abstract

1 Introduction

Definition 1.1.

Theorem 1.2.

Proposition 1.3.

Definition 1.4.

Theorem 1.5.

Theorem 1.6.

Outline of the paper.

Related work.

2 Focal-free hypergraphs

Observation 2.1.

2.1 The upper bound of fr​(n,k)f_{r}(n,k)

Theorem 2.2.

Lemma 2.3.

Proof.

Proof of Theorem 2.2.

Remark 2.4.

Proof of Proposition 1.3.

2.2 The lower bound of fr​(n,k)f_{r}(n,k)

Theorem 2.5.

Definition 2.6 (Packings and induced packings).

Lemma 2.7 ([15]).

Proof of Theorem 2.5.

Observation 2.8.

3 Focal-free codes

3.1 Upper and lower bounds of frq​(n)f_{r}^{q}(n)

Observation 3.1.

Theorem 3.2.

Remark 3.3.

Theorem 3.4.

3.2 Exact values of frq​(n)f_{r}^{q}(n)

Theorem 3.5.

Lemma 3.6.

Proof.

Lemma 3.7 (see [7, Theorems III.7.18 and III.7.20, page 226]).

Proof of Theorem 1.6.

Proof of Theorem 3.5.

4 Concluding remarks

Acknowledgements

References

Appendix A Proof of Theorem 1.5

Definition A.1.

A.1 Proof of Theorem 3.2

Lemma A.2.

Proof.

Proof of Theorem 3.2.

A.2 Proof of Theorem 3.4

Lemma A.3 ([22]).

Proof of Theorem 3.4.

2.1 The upper bound of $f_{r}(n,k)$

2.2 The lower bound of $f_{r}(n,k)$

3.1 Upper and lower bounds of $f_{r}^{q}(n)$

3.2 Exact values of $f_{r}^{q}(n)$