Size and structure of large $(s,t)$-union intersecting families

TL;DR

This paper characterizes the size and structure of large $(s,t)$-union intersecting families of sets, extending classical theorems like Erd{"H}os-Ko-Rado and confirming conjectures, with applications to Kneser graphs and stability results.

Contribution

It provides a detailed characterization of maximum $(s,t)$-union intersecting families and extends stability results, generalizing recent theorems and conjectures in combinatorics.

Findings

Characterization of maximum $(s,t)$-union intersecting families for large $n$.

Proof of a Hilton-Milner-type stability theorem for $(1,t)$-union intersecting families.

Extension of results to $K_{s_1, dots,s_{r+1}}$-free subgraphs of Kneser graphs.

Abstract

A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set $X$. An $(s,t)$-union intersecting family is a family of $k$-sets on an $n$-set $X$ such that for any $A_1,\ldots,A_{s+t}$ in this family, $\left(\cup_{i=1}^sA_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ Let $\ell(\F)$ be the minimum number of sets in $\F$ such that by removing them the resulting subfamily is intersecting. In this paper, for sufficiently large $n$, we characterize the size and structure of $(s,t)$-union intersecting families with maximum possible size and $\ell(\F)\geq s+\beta$. This allows us to find out the size and structure of some large and maximal $(s,t)$-union intersecting families. Our results are nontrivial extensions of some recent generalizations of the Erd{\H o}s-Ko-Rado theorem such as the Han and Kohayakawa theorem 2017 which finds the structure of the third largest intersecting family, the Kostochka and Mubayi theorem 2017, and the more recent Kupavskii's theorem 2018 whose both results determine the size and structure of the $i$th largest intersecting family of $k$-sets for $i\leq k+1$. In particular, we prove that a Hilton-Milner-type stability theorem holds for $(1,t)$-union intersecting families, that indeed, confirms a conjecture of Alishahi and Taherkhani 2018. We extend our results to $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In fact, when $n$ is sufficiently large, we characterize the size and structure of large and maximal $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In particular, when $s_1=\cdots=s_{r+1}=1$ our result provides some stability results related to the famous Erd{\H o}s matching conjecture.

Equations

$$\operatorname{\mathcal{HM}}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{A|\,1\in A,\,\,A\cap B\neq\varnothing\}\cup\{B\}$$

$$\operatorname{\mathcal{HM}}^{\prime}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{A|\,|A\cap\{1,2,3\}|\geq 2\}.$$

$$m\leq\left\{\begin{array}[]{lll}{n-1\choose k-1}-{n-k\choose k-1}+n-k&&\,\,{\rm if}\,\,2<k\leq s+2,\\ &&\\ {n-1\choose k-1}-{n-k\choose k-1}+{n-k-s\choose k-s-1}+s&&\,\,{\rm if}\,\,k\leq 2\,\,{\rm or}\,\,k\geq s+2.\end{array}\right.$$

$$\operatorname{\mathcal{J}}_{i}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{A:E\subset A,\,\,A\cap J\neq\varnothing\}\cup\{A:J\subset A\}\cup\{A:1\in A,\,\,A\cap E\neq\varnothing\}.$$

$$\operatorname{\mathcal{F}}_{i}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}[2,k+1]\cup[i+1,k+i]\cup\{A:1\in A,A\cap[2,k+i]=\varnothing\}.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-\chi (G)+1}{k}\right)+\eta (G)-1.$$

$$|F\setminus (\bigcup _{i\in L}{S}_i)|=\eta (G)-1$$

$$\left(\bigcup _{i=1}^s{A}_i\right)\cap \left(\bigcup _{i=1}^t{A}_{s+i}\right)\ne \varnothing .$$

$$S_1\cup \{F_j|1\le j\le s-1,1\notin F_j\}$$

$$\operatorname{\mathcal{F}}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\operatorname{\mathcal{S}}_{1}\cup\{A_{i}|\,1\leq i\leq s-1,\,1\notin A_{i}\}$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-sk-1}{k-1}\right)+s+t-1.$$

$$\operatorname{\mathcal{F}}_{s,t}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{A:1\in A,A\cap[2,sk+1]\neq\varnothing\}\cup\{A_{1},\dots,A_{s}\}\cup\{F_{1},\dots,F_{t-1}\}$$

$$T(A_{1},\ldots,A_{s+\beta}:s)\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{x|\,\text{there exist distinct}\,\,i_{1},i_{2},\ldots,i_{\beta+1}\,\text{such that}\,\,x\in\cap_{j=1}^{\beta+1}A_{i_{j}}\}.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-\lfloor \frac{(s+\beta )k}{\beta +1}\rfloor -1}{k-1}\right)+s+t+\hat{\beta }-1.$$

$$\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k-1}{k-1}\right)+t.$$

$$\operatorname{\mathcal{J}}^{1,t}_{i}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{i}:1)\cup\{A_{1},\ldots,A_{i}\}\cup\operatorname{\mathcal{B}}_{1}\cup\cdots\cup\operatorname{\mathcal{B}}_{i},$$

$$\operatorname{\mathcal{B}}_{j}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{B_{p}:p\in[t-1],B_{p}\cap E=\varnothing,\,J\setminus B_{p}=\{x_{j}\}\}$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-k-\gamma }{k-\gamma -1}\right)+\gamma t.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-r}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-s_{r+1}k-r}{k-1}\right)+s_r+s_{r+1}-1.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-r}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-\lfloor \frac{(s_{r+1}+\beta )k}{\beta +1}\rfloor -r}{k-1}\right)+s_r+s_{r+1}+\hat{\beta }-1.$$

$$\bigcup _{i=1}^{r-1}{S}_i\cup S_r^{[r-1]}(A_1,\dots ,A_{s_{r+1}+\hat{\beta }}:s_{r+1})\cup \{A_1,\dots ,A_{s_{r+1}+\hat{\beta }}\}\cup \{F_1,\dots ,F_{s_r-1}\}$$

$$\operatorname{\mathcal{J}}^{1,t}_{i,r}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},\ldots,A_{i}:1)\cup\operatorname{\mathcal{B}}_{1}\cup\cdots\cup\operatorname{\mathcal{B}}_{i}$$

$$\operatorname{\mathcal{B}}_{j}\mbox{$\ \stackrel{{\scriptstyle\rm def}}{{=}}\ $}\{B_{p}:p\in[t-1],B_{p}\cap E=\varnothing,\,J\setminus B_{p}=\{1,\ldots,r-1,x_{j}\}\}.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-r}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-k-r+1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-k-r-\gamma +1}{k-\gamma -1}\right)+\gamma t.$$

$$|F|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-\lfloor \frac{(s+1)k}{2}\rfloor -1}{k-1}\right)+(s+1)t.$$

$$|T(A_1,\dots ,A_{s+\beta }:s)|=\lfloor \frac{(s+\beta )k}{\beta +1}\rfloor .$$

$$|S_1(A_1,\dots ,A_{s+\beta }:s)|=\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-|T(A_1,\dots ,A_{s+\beta }:s)|-1}{k-1}\right)+O(n^{k-3}).$$

$$|S_1(A_1,\dots ,A_{1+\beta }:1)|=\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-|T(A_1,\dots ,A_{1+\beta }:1)|-1}{k-1}\right)+O(n^{k-3}).$$

Full text

As the induced subgraph on an $(s,t)$-union intersecting family in the Kneser graph $\operatorname{\operatorname{KG}}_{n,k}$ is a $K_{s,t}$-free subgraph, we can extend our results to $K_{s_{1},\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In fact, when $n$ is sufficiently large, we characterize the size and structure of large and maximal $K_{s_{1},\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In particular, when $s_{1}=\cdots=s_{r+1}=1$ our result provides some stability results related to the famous Erdős matching conjecture.

1. Introduction and Main Result

1.1. Erdős-Ko-Rado theorem and its generalization

Let $n$ and $k$ be two positive integers such that $n\geq k$. The symbol $[n]$ stands for the set $\{1,\ldots,n\}$ and the symbol $[k,n]$ stands for the set $[n]\setminus[k-1]$. The family of all $k$-element subsets (or $k$-sets) of $[n]$ is denoted by $[n]\choose k$. In this paper, we only consider families which consist of $k$-sets on $[n]$. A family $\operatorname{\mathcal{F}}$ is said to be intersecting if the intersection of every two members of $\operatorname{\mathcal{F}}$ is non-empty. If all members of $\operatorname{\mathcal{F}}$ contain a fixed element of $[n]$, then it is clear that $\operatorname{\mathcal{F}}$ is an intersecting family which is called a star or a trivial family. For each $i\in[n]$, the family \operatorname{\mathcal{S}}_{i}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }\{A\in{[n]\choose k}|i\in A\} is a maximal star. Also, the following two families are well-known examples for intersecting families. Let $B$ be a $k$-set of $[n]$ such that $1\not\in B$. Define

[TABLE]

and

[TABLE]

Note that for $2\leq k\leq 3$, we have $|\operatorname{\mathcal{HM}}|=|\operatorname{\mathcal{HM}}^{\prime}|$ and if $n>2k$ and $k\geq 4$, then $|\operatorname{\mathcal{HM}}|>|\operatorname{\mathcal{HM}}^{\prime}|$.

The well-known Erdős-Ko-Rado theorem [9] states that every intersecting family of ${[n]\choose k}$ has cardinality at most ${n-1\choose k-1}$ provided that $n\geq 2k$; moreover, if $n>2k$, then the only intersecting families of this cardinality are maximal stars.

As a generalization of the Erdős-Ko-Rado theorem, Hilton and Milner [24] proved a useful and interesting stability result. They showed that for $n>2k$ the maximum possible size of a nontrivial intersecting family $\operatorname{\mathcal{F}}$ of ${[n]\choose k}$ is ${n-1\choose k-1}-{n-k-1\choose k-1}+1$. Furthermore, equality is possible only for a family $\operatorname{\mathcal{F}}$ which is isomorphic to $\operatorname{\mathcal{HM}}$ or $\operatorname{\mathcal{HM}}^{\prime}$, the latter can hold only for $k\leq 3$.

A family $\operatorname{\mathcal{F}}$ is called a Hilton-Milner family if $\operatorname{\mathcal{F}}$ is isomorphic to a subfamily of $\operatorname{\mathcal{HM}}$ for some $k$ or it is isomorphic to a subfamily of $\operatorname{\mathcal{HM}}^{\prime}$ for $k\in\{2,3\}$.

There also exist some other interesting extensions of Erdős-Ko-Rado and Hilton-Milner theorems in the literature ( e.g.[1, 2, 5, 6, 12, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 31, 32, 33, 35]).

The Erdős-Ko-Rado theorem determines the maximum size of an intersecting family of $k$-sets on $[n]$ and the Hilton-Milner theorem shows that a nontrivial intersecting family has cardinality at most ${n-1\choose k-1}-{n-k-1\choose k-1}+1.$ Beyond Hilton-Milner theorem, it was shown by Hilton and Milner [24] that the maximum size of a nontrivial intersecting family which is not a Hilton-Milner family is at most ${n-1\choose k-1}-{n-k-1\choose k-1}-{n-k-2\choose k-2}+2$. In fact they proved the following interesting result (see [23, 24]).

Theorem A**.**

[24]* Let $n,k$, and $s$ be positive integers with $\min\{3,s\}\leq k\leq{n\over 2}$ and let $\operatorname{\mathcal{F}}=\{A_{1},\ldots,A_{m}\}$ be an intersecting family on $[n]$. If for any $S\subset[m]$ with $|S|>m-s$, we have $\cap_{i\in S}A_{i}=\varnothing$, then*

[TABLE]

Moreover, the bounds in Inequality $\rm({\ref{eq:hm}})$ are the best possible.

Recently, Han and Kohayakawa gave a different and simpler proof of Theorem A. Moreover, they characterized all extremal families achieving the bounds in (1) (for more details see [23]). In this regard they introduced the following construction.

Definition 1**.**

Let $i$ be a nonnegative integer. For any $(i+1)$-set $J\subset[n]$ with $1\in J$ and any $(k-1)$-set $E\subset[n]\setminus J$, define the family $\operatorname{\mathcal{J}}_{i}$ as follows,

[TABLE]

Note that $\operatorname{\mathcal{J}}_{0}=\operatorname{\mathcal{S}}_{1}$, $\operatorname{\mathcal{J}}_{1}=\operatorname{\mathcal{HM}}$, $|\operatorname{\mathcal{J}}_{i}|={n-1\choose k-1}-{n-k\choose k-1}+{n-k-i\choose k-i-1}+i$, and $|\operatorname{\mathcal{J}}_{i}\setminus\operatorname{\mathcal{S}}_{1}|=i$.

Theorem B**.**

[23]* Let $n,k$ be positive integers with $3\leq k<{n\over 2}$ and let $\operatorname{\mathcal{F}}$ be an intersecting family of $k$-sets on $[n]$. Assume that $\operatorname{\mathcal{F}}$ is neither a star nor a Hilton-Milner family. Then $|\operatorname{\mathcal{F}}|\leq|\operatorname{\mathcal{J}}_{2}|$. Moreover, for $k\geq 5$, equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{J}}_{2}$.*

Definition 2**.**

For $i\leq k$ let us define the family $\operatorname{\mathcal{F}}_{i}$ of ${[n]\choose k}$ as follows,

[TABLE]

In [29], Kostochka and Mubayi proved that the size of an intersecting family which is neither a star nor is contained in $\operatorname{\mathcal{J}}_{i}$, for $i\in\{1,\ldots,k-1,n-k\}$, is at most $|\operatorname{\mathcal{F}}_{3}|$ for $k\geq 5$ and sufficiently large $n=n(k)$. Also, more recently Kupavskii [32] extended this result and showed that the same result holds when $5\leq k<{n\over 2}$.

Theorem C**.**

[29, 32]* Let $n,k$ be positive integers with $5\leq k<{n\over 2}$ and let $\operatorname{\mathcal{F}}$ be an intersecting family of $k$-sets on $[n]$ with $|\operatorname{\mathcal{F}}|>|\operatorname{\mathcal{F}}_{3}|$. Then $\operatorname{\mathcal{F}}\subseteq\operatorname{\mathcal{J}}_{i}$ for $i\in\{0,1,\ldots k-1,n-k\}$.*

1.2. $G$-free subgraphs of Kneser graphs and $(s,t)$-union intersecting families

Let $n\geq 2k$. The Kneser graph $\operatorname{\operatorname{KG}}_{n,k}$ is a graph whose vertex set is ${[n]\choose k}$ where two vertices are adjacent if their corresponding sets are disjoint. From another point of view, the Erdős-Ko-Rado theorem [9] determines the maximum independent sets of Kneser graphs. Recalling the fact that an independent set in a graph $G$ is a subset of vertices containing no subgraph isomorphic to $K_{2}$, the following question was asked in [1].

“Given a graph $G$, how large a family $\operatorname{\mathcal{F}}\subseteq{[n]\choose k}$ must be chosen to guarantee that $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ has some subgraph isomorphic to $G$? What is the structure of the largest subset $\operatorname{\mathcal{F}}\subseteq{[n]\choose k}$ for which $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ has no subgraph isomorphic to $G$?”

This problem has already been investigated for some special cases. In particular, if $G=K_{2}$, the answer is the Erdős-Ko-Rado theorem and if $G=K_{1,t}$ or $G=K_{s,t}$, the question has been studied in [1, 20] and [1, 28], respectively.

If $G=K_{r+1}$, the question is equivalent to the famous Erdős matching conjecture [7] where it has been studied extensively in the literature (e.g. [8, 16, 34, 4, 7, 19, 25]). As one of the strongest results in this regard, Frankl [15] has confirmed the Erdős matching conjecture for $n\geq(2r+1)k-r$.

Also, this problem can be considered as a vertex Turán problem as follows. Given a host graph $H$ (which is the Kneser graph in our case) and a forbidden graph $G$, what is the size and structure of the largest set $U\subseteq V(H)$ such that the induced subgraph $H[U]$ is $G$-free?

In[1], Alishahi and the author determined the size and structure of a family $\operatorname{\mathcal{F}}$ on $[n]$ with maximum size such that the induced subgraph $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $G$-free provided that $n$ is sufficiently large.

In the sequel, a subgraph $H$ of a given graph $G$ is called a special subgraph if removing its vertices from $G$ reduces the chromatic number by one.

Theorem D**.**

[1]* Let $k\geq 2$ be a fixed positive integer and $G$ be a fixed graph for which $\chi(G)$ is the chromatic number and $\eta(G)$ is the minimum possible size of a color class of $G$ over all possible proper $\chi(G)$-colorings of $G$. There exists a threshold $N(G,k)$ such that for any $n\geq N(G,k)$ and for any $\operatorname{\mathcal{F}}\subseteq{[n]\choose k}$, if $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ has no subgraph isomorphic to $G$, then*

[TABLE]

Moreover, equality holds if and only if there is a $(\chi(G)-1)$-set $L\subseteq[n]$ such that

[TABLE]

and $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}\setminus(\bigcup_{i\in L}\operatorname{\mathcal{S}}_{i})]$ has no subgraph isomorphic to a special subgraph of $G$.

Let $s$ and $t$ be two positive integers such that $t\geq s$. A family of $k$-sets $\operatorname{\mathcal{F}}$ on $[n]$ is said to be an $(s,t)$-union intersecting family if for any subfamily $\{A_{1},A_{2},\ldots,A_{s+t}\}$ of $\operatorname{\mathcal{F}}$,

[TABLE]

It is straight forward to see that a family $\operatorname{\mathcal{F}}$ is an $(s,t)$-union intersecting family if and only if $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{s,t}$-free. As a generalization of the Erdős-Ko-Rado theorem in [28] Katona and Nagy showed that for sufficiently large $n$, any $(s,t)$-union intersecting family has cardinality at most ${n-1\choose k-1}+s-1$. Alishahi and the author improved this result, and moreover, characterized the extremal cases in [1]. Also, in [1] an asymptotic Hilton-Milner-type stability theorem was proved for an $(s,t)$-union intersecting family on [n]. More recently, an explicit extension of this result is proved by Grebner, Methuku, Nagy, Patkós, and Vizer [21]. They show that for $2\leq s\leq t$, the size of an $(s,t)$-union intersecting family on [n], which is not isomorphic to a subfamily of

[TABLE]

for some $F_{1},\ldots,F_{s-1}$, is at most ${n-1\choose k-1}-{n-sk-1\choose k-1}+s+t-1$ and characterize the largest one. In fact, they prove that a Hilton-Milner-type theorem for an $(s,t)$-union intersecting family is true when $t\geq s\geq 2$ and $n$ is sufficiently large.

Note that the first largest $(s,t)$-union intersecting family is the union of the star $\operatorname{\mathcal{S}}_{1}$ and $s-1$ other $k$-sets. For $i\geq 2$, we say $\operatorname{\mathcal{F}}$ is the $i$th largest $(s,t)$-union intersecting family, if $\operatorname{\mathcal{F}}$ is a maximal $(s,t)$-union intersecting subfamily of $[n]\choose k$ and is not contained in the $j$th largest $(s,t)$-union intersecting family for every $j\leq i-1$. Indeed, the Hilton-Milner theorem determines the size and structure of the second $(1,1)$-union intersecting family. Also, Han and Kohayakawa in [23] characterize the size and structure of the third $(1,1)$-union intersecting family. For sufficiently large $n$, Kostochka and Mubayi in [29] and Kupavskii in [32] find the size and structure of the $i$th $(1,1)$-union intersecting family when $i\leq k+1$. In this regard, for sufficiently large $n$, Grebner et al. in [21] determine the size and structure of the second largest $(s,t)$-union intersecting family when $t\geq s\geq 2$. Motivated by the mentioned results, one may naturally ask the following questions.

For a family $\operatorname{\mathcal{F}}$ and an integer $r\geq 2$, let $\ell_{r}(\operatorname{\mathcal{F}})$ denote the minimum number $m$ such that by removing $m$ sets from $\operatorname{\mathcal{F}}$, the resulting family has no $r$ pairwise disjoint sets. For simplicity of notation, let \ell(\operatorname{\mathcal{F}})\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }\ell_{2}(\operatorname{\mathcal{F}}).

Question 1**.**

What are the size and structure of the $i$th largest $(s,t)$-union intersecting family?

Question 2**.**

What are the size and structure of the largest $(s,t)$-union intersecting family with $\ell(\operatorname{\mathcal{F}})\geq s+\beta$?

It is worth mentioning that each family $\operatorname{\mathcal{F}}$ with $\ell(\operatorname{\mathcal{F}})=s-1$ is $(s,t)$-union intersecting and the largest $(s,t)$-union intersecting family

[TABLE]

has $\ell(\operatorname{\mathcal{F}})=s-1$. Grebner et al. in [21], as their main result, determine the size and structure of the largest $(s,t)$-union intersecting family with $\ell(\operatorname{\mathcal{F}})\geq s$, when $t\geq s\geq 2$ and $n$ is sufficiently large. By using the Hilton-Milner theorem and their result, one can verify that the second largest $(s,t)$-union intersecting family must have $\ell(F)\geq s$. In fact, the next theorem determines the second largest $(s,t)$-union intersecting family.

Theorem E**.**

[21]* For any $2\leq s\leq t$ and $k$ there exists $N=N(s,t,k)$ such that if $n\geq N$ and $\operatorname{\mathcal{F}}$ is a family with $\ell(\operatorname{\mathcal{F}})\geq s$ and $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{s,t}$-free, then we have*

[TABLE]

Moreover, equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to some $\operatorname{\mathcal{F}}_{s,t}$ which is defined as follows,

[TABLE]

where A_{i}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }[(i-1)k+2,ik+1] for each $1\leq i\leq s$, and for each $j\leq t-1$, we have $1\in F_{j}$ and $F_{j}\cap[2,sk+1]=\varnothing$.

Motivated by the mentioned results and questions, in this paper, we try to determine the structure and size of an $(s,t)$-union intersecting family with maximum size when $\ell(\operatorname{\mathcal{F}})\geq s+\beta$ and $n$ is sufficiently large.

To state our main results, we need the following definitions.

Definition 3**.**

Let $n,k,s,$ and $\beta$ be fixed nonnegative integers. Let $A_{1},\ldots,A_{s+\beta}$ be $s+\beta$ pairwise distinct $k$-sets on $[n]$ such that $1\not\in\cup_{i=1}^{s+\beta}A_{i}$. Define $\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)$ as the largest subfamily of $\operatorname{\mathcal{S}}_{1}$ such that each $A\in\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)$ is disjoint from at most $s-1$ of $A_{i}$s. Also, define

[TABLE]

Note that when $\beta=0$, we have $T(A_{1},\ldots,A_{s}:s)=\cup_{i=1}^{s}A_{i}$ and $\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s}:s)$ is equal to $\operatorname{\mathcal{S}}_{1}\setminus\{A:A\cap(\cup_{i=1}^{s}A_{i})=\varnothing\}$. Also, when $s=1$ the family $\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{1+\beta}:1)$ is equal to $\operatorname{\mathcal{S}}_{1}\setminus\{A|\,A\cap A_{i}=\varnothing\,\,\text{for some }\,1\leq i\leq\beta+1\}$ and $T(A_{1},\ldots,A_{1+\beta}:1)=\cap_{i=1}^{1+\beta}A_{i}$.

Definition 4**.**

Let $k,s,$ and $\beta$ be fixed nonnegative integers. If $\lfloor{(s+{\beta})k\over{\beta}+1}\rfloor>k$, define \hat{\beta}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }\hat{\beta}(k,s,\beta) as the largest positive integer such that $\lfloor{(s+{\beta})k\over{\beta}+1}\rfloor=\lfloor{(s+\hat{\beta})k\over\hat{\beta}+1}\rfloor$; else if $\lfloor{(s+{\beta})k\over{\beta}+1}\rfloor=k$, define \hat{\beta}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }\beta.

Now, we are in a position to state our first result.

Theorem 1**.**

Let $k\geq 3,t\geq s\geq 1$, and $\beta$ be fixed nonnegative integers and $n=n(s,t,k,\beta)$ be sufficiently large. Let $\operatorname{\mathcal{F}}$ be an $(s,t)$-union intersecting family such that $\ell(\operatorname{\mathcal{F}})\geq s+\beta$. Then

[TABLE]

Equality holds if and only if there exist pairwise distinct $k$-sets $A_{1},\ldots,A_{s+\hat{\beta}}$ and $F_{1},\ldots,F_{t-1}$ such that

1. (1)

$1\notin\bigcup\limits_{i=1}^{s+\hat{\beta}}A_{i}$, 2. (2)

$|T(A_{1},\ldots,A_{s+\hat{\beta}}:s)|=\lfloor{(s+\beta)k\over\beta+1}\rfloor$, 3. (3)

for each $i\leq t-1$, $F_{i}\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\hat{\beta}}:s),$ and 4. (4)

the family $\{A_{1},\ldots,A_{s+\hat{\beta}},F_{1},\ldots,F_{t-1}\}$ is an $(s,t)$-union intersecting family

and $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\hat{\beta}}:s)\cup\{A_{1},\ldots,A_{s+\hat{\beta}}\}\cup\{F_{1},\ldots,F_{t-1}\}.$

It is worth mentioning that Theorem E follows from Theorem 1 by choosing $\beta=0$ and $s\geq 2$. By applying the previous theorem and using some properties of $T(A_{1},\ldots,A_{s+{\beta}}:s)$, we can find out the $j$th largest $(s,t)$-union intersecting family for some $j$’s. We provide a more detailed analysis in our remarks proceeding the proof of Theorem 1.

Note that perhaps for some $k,s,$ and $\beta$ there exist no distinct pairwise $A_{1},\ldots,A_{s+\beta}$ satisfying Condition (2) in the previous theorem. For example, one may choose $k=3,s=3$, and $\beta=5$. Thus, we have $\lfloor{(s+\beta)k\over\beta+1}\rfloor=4$. Since $\cup_{i=1}^{8}A_{i}=T(A_{1},\ldots,A_{8}:3)$, if there exist $A_{1},\ldots,A_{8}$ for which $|T(A_{1},\ldots,A_{8}:3)|=4$, then at least two of $A_{i}$’s must be identical, which is not possible. Therefore, for some $k,s,$ and $\beta$ there do not exist any $A_{1},\ldots,A_{s+\beta}$ such that $|T(A_{1},\ldots,A_{s+{\beta}}:s)|=\lfloor{(s+\beta)k\over\beta+1}\rfloor$. Consequently, as we will show in the proof of Theorem 1, each $(s,t)$-union intersecting family $\operatorname{\mathcal{F}}$ is of size less than ${n-1\choose k-1}-{n-\lfloor{(s+{\beta})k\over{\beta}+1}\rfloor-1\choose k-1}$ showing that if $\ell(\operatorname{\mathcal{F}})\geq s+\beta$, then $|\operatorname{\mathcal{F}}|$ is at most ${n-1\choose k-1}-{n-|T(A_{1},\ldots,A_{s+{\beta}}:s)|-1\choose k-1}+O(n^{k-3})$.

When $\operatorname{\mathcal{F}}$ is a $(1,t)$-union intersecting family of ${[n]\choose k}$ (or $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is a $K_{1,t}$-free subgraph of $\operatorname{\operatorname{KG}}_{n,k}$) it is proved that every $(1,t)$-union intersecting family with at least ${n-1\choose k-1}-{n-k-1\choose k-1}+(t-1){2k-1\choose k-1}+t$ members is contained in some star $\operatorname{\mathcal{S}}_{i}$ for sufficiently large $n$ [1]. Moreover, it is posed as a conjecture in the same refrence that for sufficiently large $n$ one can replace the term $(t-1){2k-1\choose k-1}$ by $1$. Note that this conjecture is an extension of the Hilton-Milner theorem. Also, if the statement of Theorem E is true for $s=1$, then the conjecture holds, however, the condition $s\geq 2$ is necessary in the proof of Theorem E presented in [21]. This conjecture is one of our motivations for this study, in which we show that the conjecture follows from Theorem 1 choosing $s=1$ and $\beta=0$. For further reference we state this fact in the following corollary.

Corollary 1**.**

Let $k\geq 3$ and $t$ be positive integers and $n=n(k,t)$ is sufficiently large. Any $(1,t)$-union intersecting family $\operatorname{\mathcal{F}}\subseteq{[n]\choose k}$, which is not contained in any star, has cardinality at most

[TABLE]

Equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to \operatorname{\mathcal{J}}_{1}^{1,t}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }\operatorname{\mathcal{S}}_{1}(A_{1}:1)\cup\{A_{1}\}\cup\{F_{1},\ldots,F_{t-1}\} where $F_{i}\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1}:1)$ for each $i\leq t-1$.

Note that if $F\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1}:1)$, it means that $1\in F$ and $F\cap A_{1}=\varnothing$. Also, note that $\operatorname{\mathcal{J}}_{1}^{1,1}$ is isomorphic to $\operatorname{\mathcal{HM}}$ and $\operatorname{\mathcal{J}}_{1}$.

Concerning our next result when $s=1,t\geq 1$, and $\beta\leq k-3$, motivated by Theorems B and C and the mentioned conjecture, we determine the maximum size and structure of a $(1,t)$-union intersecting family $\operatorname{\mathcal{F}}$ with $\ell(\operatorname{\mathcal{F}})\geq 1+\beta$. Note that when $s=1$ and $\beta\geq 1$, Theorem 1 does not give a sharp bound for maximum size of $(1,t)$-union intersecting families. This result leads us to determine the $i$th largest $(1,t)$-union intersecting families where $i\leq k-2$.

Before stating the next result we need to introduce the following construction.

Definition 5**.**

Let $i\leq k-1$ be a nonnegative integer. For any $(i+1)$-set $J=\{1,x_{1},\ldots,x_{i}\}$ of $[n]$ and any $(k-1)$-set $E\subset[n]\setminus J$. Let $A_{1},\ldots,A_{i}$ be $i$ $k$-subsets on $[n]\setminus\{1\}$ such that $\cap_{j=1}^{i}A_{j}=E$ and $A_{j}\setminus E=\{x_{j}\}$ for each $j\leq i$ define $\operatorname{\mathcal{J}}^{1,t}_{i}$ as follows

[TABLE]

where $\operatorname{\mathcal{B}}_{j}$, for $j\leq i$, defined as follows

[TABLE]

Notice that $\operatorname{\mathcal{J}}^{1,1}_{i}$ isomorphic to $\operatorname{\mathcal{J}}_{i}$ and $\operatorname{\mathcal{J}}_{i}=\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{i}:1)\cup\{A_{1},\ldots,A_{i}\}$. Since $\operatorname{\mathcal{B}}_{j}$’s in the definition of $\operatorname{\mathcal{J}}^{1,t}_{i}$ are pairwise disjoint. Therefore, $|\operatorname{\mathcal{J}}^{1,t}_{i}|=|\operatorname{\mathcal{J}}_{i}|+i(t-1)$.

For $s=1$ we can state a strong improvement of Corollary 1 and Theorem 1 as follows.

Theorem 2**.**

Let $k\geq 5,t\geq 1$, and $\gamma=1+\beta\leq k-2$ be nonnegative integers and $n=n(k,t,\gamma)$ is sufficiently large. Let $\operatorname{\mathcal{F}}$ be a $(1,t)$-union intersecting family with $\ell(\operatorname{\mathcal{F}})\geq\gamma$. Then

[TABLE]

Equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{J}}_{\gamma}^{1,t}$.

It can be seen that the next corollary is a direct consequence of Theorem 2. Notice that we need to apply Theorem C to prove it.

Corollary 2**.**

Let $n$, $k\geq 5$, $t\geq 1$, and $\gamma\leq k-2$ be nonnegative integers such that $n=n(k,t,\gamma)$ is sufficiently large. Let $\operatorname{\mathcal{F}}$ be a $(1,t)$-union intersecting family that is not isomorphic to a subfamily of $\operatorname{\mathcal{J}}_{i}\cup\operatorname{\mathcal{B}}$ where $\operatorname{\mathcal{B}}\subseteq\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{J}}_{i}$ and $0\leq i\leq\gamma-1$. Then

[TABLE]

Equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to some $\operatorname{\mathcal{J}}_{\gamma}^{1,t}$.

Note that if we choose $\operatorname{\mathcal{B}}=\operatorname{\mathcal{B}}_{1}\cup\ldots\cup\operatorname{\mathcal{B}}_{i}$, where $\operatorname{\mathcal{B}}_{j}$’s come from Definition 5, then $\operatorname{\mathcal{J}}_{i}\cup\operatorname{\mathcal{B}}=\operatorname{\mathcal{J}}_{i}^{1,t}$.

1.3. Some stability results for the Erdős matching conjecture and its generalization

The Erdős matching conjecture is one of the famous open problems in extremal set theory. It states that for $n\geq(r+1)k$, the size of the largest subset $\operatorname{\mathcal{F}}\subseteq{[n]\choose k}$ for which $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ has no copy of $K_{r+1}$ is $\max\{{(r+1)k-1\choose k},{n\choose k}-{n-r\choose k}\}$. In recent years, this conjecture has received considerable attention. It has been already proved that the conjecture is true for $k\leq 3$ (see [8, 16, 34]). Also, improving the earlier results in [4, 7, 19, 25], Frankl [15] confirmed the conjecture for $n\geq(2r+1)k-r$; moreover, he determined the structure of the extremal cases in this range. Frankl and Kupavskii [18] proved a Hilton-Milner-type stability theorem for the Erdős matching conjecture for $n\geq(2+o_{r}(1))(r+1)k$ as a significant improvement of a classical result due to Bollobás, Daykin and Erdős [4].

Hereafter, we will focus on complete multipartite graphs $K_{s_{1},s_{2},\cdots,s_{r+1}}$ as a forbidden subgraph. We show that the previous results for $(s,t)$-union intersecting family can be extended to $K_{s_{1},s_{2},\cdots,s_{r+1}}$-free subgraph of Kneser graphs instead of $K_{s,t}$-free subgraphs of Kneser graphs as nontrivial extensions of the Erdős matching conjecture. In this regard, Grebner et. al. show that a generalization of Theorem E holds when $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{s_{1},s_{2},\cdots,s_{r+1}}$-free when $s_{1}\geq\cdots\geq s_{r+1}\geq 2$. They determine the size and structure of the second largest family $\operatorname{\mathcal{F}}$ on $[n]$ such that $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{s_{1},s_{2},\ldots,s_{r+1}}$-free, where $s_{r+1}\geq 2$ for sufficiently large $n$. Before stating their result, we need an extension of the construction of Definition 3.

Definition 6**.**

Let $n,k,s,$ and $\beta$ be fixed positive integers. Let $A_{1},\ldots,A_{s+\beta}$ be $s+\beta$ pairwise distinct $k$-sets on $[n]$ such that $[r]\bigcap(\cup_{i=1}^{s+\beta}A_{i})=\varnothing$. Define $\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},\ldots,A_{s+\beta}:s)$ as the largest subfamily of $\operatorname{\mathcal{S}}_{r}$ such that each $A\in\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},\ldots,A_{s+\beta}:s)$ is disjoint from $[r-1]$ and at most $s-1$ of $A_{i}$’s.

Note that the family $\operatorname{\mathcal{S}}_{r}(A_{1},\ldots,A_{s+\beta}:s)$ in Definition 3 is a special case of Definition 6 when $r=1$.

Theorem F**.**

[21]* For any $k\geq 2$ and integers $s_{1}\geq s_{2}\cdots\geq s_{r+1}\geq 2$ there exists $N=N(s_{1},s_{2},\dots,s_{r+1},k)$ such that if $n\geq N$ and $\operatorname{\mathcal{F}}$ is a family with $\ell_{r+1}(F)\geq s_{r+1}$ and $\operatorname{\operatorname{KG}}_{n,k}[F]$ is $K_{s_{1},s_{2},\cdots,s_{r+1}}$-free, then we have*

[TABLE]

Moreover, equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},\ldots,A_{s}:s)\cup\{A_{1},\ldots,A_{s}\}\cup\{F_{1},\ldots,F_{t-1}\}$.

We are able to prove an analog of the previous theorem by using the Erdős-Stone-Simonovits theorem and Theorem 1.

Theorem 3**.**

Let $k\geq 3,s_{1}\geq\cdots\geq s_{r+1}\geq 1$ and $\beta$ be fixed nonnegative integers and $n=n(s_{1},\ldots,s_{r+1},k,\beta)$ is sufficiently large. Assume that $\hat{\beta}=\hat{\beta}(k,s_{r+1},\beta)$. Let $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ be $K_{(s_{1},\ldots,s_{r+1})}$-free such that $\ell_{r+1}(\operatorname{\mathcal{F}})\geq s_{r+1}+\beta$. Then

[TABLE]

Equality holds if and only if there exist $s_{r+1}+\hat{\beta}$ pairwise distinct $k$-sets $A_{1},\ldots,A_{s_{r+1}+\hat{\beta}}$ such that

• •

$[r]\bigcap(\bigcup\limits_{i=1}^{s_{r+1}+\hat{\beta}}A_{i})=\varnothing$,

• •

$|T(A_{1},\ldots,A_{s_{r+1}+\hat{\beta}})|=\lfloor{(s_{r+1}+\beta)k\over\beta+1}\rfloor$,

• •

for each $i\leq s_{r}-1$, $F_{i}\in\operatorname{\mathcal{S}}_{r}\setminus\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},\ldots,A_{s_{r+1}+\hat{\beta}}:s_{r+1})$ and $F_{i}\cap[r-1]=\varnothing$, and

• •

the family $\{A_{1},\ldots,A_{s_{r+1}+\hat{\beta}},F_{1},\ldots,F_{s_{r}-1}\}$ is an $(s_{r+1},s_{r})$-union intersecting family and

$\operatorname{\mathcal{F}}$* is isomorphic to*

[TABLE]

When $s_{r+1}=1$ same as Theorem 2 we are able to prove a stronger result than Theorem 3 which yields a new stability result for Erdős matching conjecture for sufficiently large $n$.

Definition 7**.**

Let $i\leq k-1$ be a nonnegative integer. For any $(i+r)$-set $J=\{1,\ldots,r,x_{1},\ldots,x_{i}\}$ of $[n]$ and any $(k-1)$-set $E\subset[n]\setminus J$. Let $A_{1},\ldots,A_{i}$ be $i$ $k$-subsets on $[n]\setminus[r]$ such that $\cap_{j=1}^{i}A_{j}=E$ and $A_{j}\setminus E=\{x_{j}\}$ for each $j\leq i$ define $\operatorname{\mathcal{J}}^{1,t}_{i,r}$ and $\operatorname{\mathcal{J}}^{\prime 1,t}_{i,r}$ as follows

[TABLE]

where $\operatorname{\mathcal{B}}_{j}$, for $j\leq i$ defined as follows,

[TABLE]

Notice that $\operatorname{\mathcal{J}}^{1,t}_{i,1}$ is isomorphic to $\operatorname{\mathcal{J}}^{1,t}_{i}$. Now we are in a position to state a stability result related to Erdős matching conjecture provided that $n$ is sufficiently large.

Theorem 4**.**

Let $k\geq 5,s_{1}\geq\cdots\geq s_{r}\geq 1$, and $\gamma(=1+\beta)\leq k-2$ be fixed nonnegative integers such that $n=n(s_{1},\ldots,s_{r},k,\gamma)$ is sufficiently large. Let $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ be $K_{(s_{1},\ldots,s_{r},1)}$-free such that $\ell_{r+1}(\operatorname{\mathcal{F}})\geq\gamma$. Then

[TABLE]

Equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{J}}^{1,s_{r}}_{\gamma,r}$

Corollary 3**.**

Let $n$, $k\geq 5$, $s_{1}\geq\cdots\geq s_{r}\geq 1$, and $\gamma(=1+\beta)\leq k-2$ be nonnegative integers such that $n=n(s_{1},\ldots,s_{r},k,\gamma)$ is sufficiently large. Let $\operatorname{\mathcal{F}}$ be a family such that $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{(s_{1},\ldots,s_{r},1)}$-free and $\operatorname{\mathcal{F}}$ is not isomorphic to a subfamily of $\operatorname{\mathcal{J}}^{1,1}_{i,r}\cup\operatorname{\mathcal{B}}$ where $\operatorname{\mathcal{B}}\subseteq\operatorname{\mathcal{S}}_{r}\setminus\operatorname{\mathcal{J}}^{1,1}_{i,r}$ and $0\leq i\leq\gamma-1$. Then

[TABLE]

Equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{J}}^{1,s_{r}}_{\gamma,r}$

2. Proofs

Before the proof of Theorem 1, let us state an interesting lemma from [21]. Here we show that a strong generalization of Lemma A is true.

Lemma A**.**

[21]* Let $s\leq t$ and let $A_{1},A_{2},\ldots,A_{s+1}$ be $k$-sets on $[n]$ such that $1\not\in\cup_{i=1}^{s+1}A_{i}$. Suppose that $\operatorname{\mathcal{F}}^{\prime}$ is a subfamily of $\operatorname{\mathcal{S}}_{1}$ such that for $\operatorname{\mathcal{F}}=\operatorname{\mathcal{F}}^{\prime}\cup\{A_{1},A_{2},\ldots,A_{s+1}\}$ the induced subgraph of $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ is $K_{s,t}$-free. There exists $n_{0}=n(k,s,t)$ such that if $n\geq n_{0}$ holds, then we have*

[TABLE]

The next lemma provides an interesting and useful generalization of Lemma A. I believe that Lemma 1 independently will be a useful result and will have more applications.

Lemma 1**.**

Let $k,s,$ and $\beta$ be fixed nonnegative integers and $n=n(k,s,\beta)$ is sufficiently large. Let $A_{1},A_{2},\ldots,A_{s+\beta}$ be $k$-sets on $[n]$ such that $1\not\in\cup_{i=1}^{s+\beta}A_{i}$. Then

1. (a)

${n-1\choose k-1}-{n-|T(A_{1},\ldots,A_{s+\beta}:s)|-1\choose k-1}\leq|\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)|.$* * 2. (b)

$|\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)|\leq{n-1\choose k-1}-{n-\lfloor{(s+\beta)k\over\beta+1}\rfloor-1\choose k-1}$* and equality holds if and only if*

[TABLE]

In particular, if $|T(A_{1},\ldots,A_{s+\beta}:s)|<\lfloor{(s+\beta)k\over\beta+1}\rfloor$, then

[TABLE] 3. (c)

For $s=1$, we have $|\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{1+\beta}:1)|\leq{n-1\choose k-1}-{n-k\choose k-1}+{n-k-\beta-1\choose k-\beta-2}.$ Moreover for $\beta\geq 1$, equality holds if and only if $|T(A_{1},\ldots,A_{1+\beta}:1)|=k-1.$

In particular, if $|T(A_{1},\ldots,A_{1+\beta}:1)|<k-1$, then

[TABLE]

Proof.

For abbreviation let $T(A_{1},\ldots,A_{s+\beta}:s)=T_{\beta}$. For the proof of Part (a), let $1\in A$. If $A\cap T_{\beta}\neq\varnothing$, then $A$ is disjoint from at most $s-1$ sets of $A_{1},A_{2},\ldots,A_{s+\beta}$. Therefore, ${n-1\choose k-1}-{n-|T_{\beta}|-1\choose k-1}\leq|\operatorname{\mathcal{S}}_{1}({A_{1},\ldots,A_{s+\beta}}:s)|.$

Now we prove Part (b). One can check that $|T_{\beta}|\leq\lfloor{(s+\beta)k\over\beta+1}\rfloor.$ Assume that $|T_{\beta}|<\lfloor{(s+\beta)k\over\beta+1}\rfloor$. Let $A\in\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)$. Therefore, $A$ intersects at least $\beta+1$ of $A_{1},\ldots,A_{s+\beta}$. We have two possibilities for $A$. Either $A\cap T_{\beta}\neq\varnothing$ or $A\cap T_{\beta}=\varnothing$ and $A$ intersects at least $\beta+1$ of $A_{1},\ldots,A_{s+\beta}$. The number of members in $\operatorname{\mathcal{S}}_{1}$ which meet $T_{\beta}$ is equal to ${n-1\choose k-1}-{n-|T_{\beta}|-1\choose k-1}$. The number of $k$-sets in $\operatorname{\mathcal{S}}_{1}$, which intersect at least $\beta+1$ of $A_{1},\ldots,A_{s+\beta}$ and have no common element with $T_{\beta}$, is at most

[TABLE]

Therefore,

[TABLE]

and then

[TABLE]

provided that $n$ is sufficiently large.

Now assume that we have the equality $|\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)|={n-1\choose k-1}-{n-\lfloor{(s+\beta)k\over\beta+1}\rfloor-1\choose k-1}.$ By contradiction assume that $|T_{\beta}|<\lfloor{(s+\beta)k\over\beta+1}\rfloor$. Using the same reasoning one may verify that when $n$ is sufficiently large, then $|\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)|$ is less than ${n-1\choose k-1}-{n-\lfloor{(s+\beta)k\over\beta+1}\rfloor-1\choose k-1}$ which is not possible.

Now suppose that $|T_{\beta}|=\lfloor{(s+\beta)k\over\beta+1}\rfloor$. To prove the last part of (b), it suffices to show that

[TABLE]

From the division algorithm, we know that $(s+\beta)k=\lfloor{(s+\beta)k\over\beta+1}\rfloor(\beta+1)+r$ where $0\leq r\leq\beta$. Since $|T_{\beta}|=\lfloor{(s+\beta)k\over\beta+1}\rfloor$, there are at most $r\leq\beta$ elements in $\cup_{i\in[s+\beta]}A_{i}$ which are not in $T_{\beta}$. Therefore, there exist $1\leq i_{1}<\ldots<i_{s}\leq s+\beta$ such that $A_{i_{1}}\cup\cdots\cup A_{i_{s}}\subseteq T_{\beta}.$ On the other hand, for every $1\leq j_{1}<\ldots<j_{s}\leq s+\beta$, we have $T_{\beta}\subseteq A_{j_{1}}\cup\cdots\cup A_{j_{s}}.$ Therefore, $T_{\beta}=A_{i_{1}}\cup\cdots\cup A_{i_{s}}.$ Assume that $1\in A$ and $A\cap T_{\beta}=\varnothing$. Hence, $A\cap(A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{s}})=\varnothing$. Therefore, $A$ is disjoint from at least $s$ $k$-subsets of $A_{1},A_{2},\ldots,A_{s+\beta}$ and consequently $A\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta}:s)$. If $1\in A$ and $A$ is disjoint from at least $s$ $k$-subsets of $A_{1},A_{2},\ldots,A_{s+\beta}$, then it is clear that each element of $A$ appears in at most $\beta$ of $A_{i}$’s and hence we have $A\cap T_{\beta}=\varnothing.$

For the proof of (c), if $|T_{\beta}|\leq k-2$, then the proof is the same as the first part of (b). Hence, we may assume that $|T_{\beta}|$ is $k-1$ or $k$. Note that when $s=1$, $T_{\beta}=\cap_{i=1}^{1+\beta}A_{i}$. If $|T_{\beta}|=k$, then $\beta$ must be equal to [math] and consequently $|\operatorname{\mathcal{S}}_{1}(A_{1}:1)|={n-1\choose k-1}-{n-k-1\choose k-1}$. Thus, we may assume that $|T_{\beta}|=|\cap_{i=1}^{1+\beta}A_{i}|=k-1$ and $\beta\geq 1$. Then, there exist $\beta+1$ elements in $[n]$, say $x_{1},\ldots,x_{\beta+1}$, such that $A_{j}\setminus T_{\beta}=\{x_{j}\}.$ Let $A\in\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{1+\beta}:1)$. Therefore, $A$ intersects each of $A_{1},\ldots,A_{1+\beta}$. We have two possibilities for $A$. Either $A\cap T_{\beta}\neq\varnothing$ or $A\cap T_{\beta}=\varnothing$ and $A$ intersects all of $A_{1},\ldots,A_{1+\beta}$. There are ${n-1\choose k-1}-{n-k\choose k-1}$ members in $\operatorname{\mathcal{S}}_{1}$ such that $A\cap T_{\beta}\neq\varnothing$. The number of $k$-sets in $\operatorname{\mathcal{S}}_{1}$, which intersect all of $A_{1},\ldots,A_{1+\beta}$ and have no common element with $T_{\beta}$, is equal ${n-k-\beta-1\choose k-\beta-2}$. Therefore,

[TABLE]

Note that when $\beta\geq k-1$, we have ${n-k-\beta-1\choose k-\beta-2}=0$. ∎

In the proof of Theorem 1 in addition to Lemma 1, we will use the following two results. The first one is a classical result on the number of edges of a $K_{s,t}$-free graph and the second one is a result on the number of disjoint pairs in a family of $k$-sets $\operatorname{\mathcal{F}}$.

Theorem G**.**

[30]* For any two positive integers $s\leq t$, if $G$ is a $K_{s,t}$-free graph with $n$ vertices, then the number of edges of $G$ is at most $({1\over 2}+o(1))(t-1)^{1\over s}n^{2-{1\over s}}$.*

Lemma B**.**

[2]* Let $\operatorname{\mathcal{F}}$ be a family $k$-sets on $[n]$. Then the number of disjoint pairs in $\operatorname{\mathcal{F}}$ is at least ${\ell(\operatorname{\mathcal{F}})^{2}\over 2{2k\choose k}}.$*

For an intersecting family $\operatorname{\mathcal{F}}^{\prime}$ on $[n]$, define $\Delta(\operatorname{\mathcal{F}}^{\prime})=\max_{i\in[n]}|\operatorname{\mathcal{F}}^{\prime}\cap\operatorname{\mathcal{S}}_{i}|$.

Proof of Theorem 1.

Let $\operatorname{\mathcal{F}}$ be an $(s,t)$-union intersecting family of ${[n]\choose k}$ with $\ell(\operatorname{\mathcal{F}})\geq s+\beta$ and cardinality

[TABLE]

We consider the following three cases.

1. (i)

$\ell(\operatorname{\mathcal{F}})=s+\beta^{\prime}$* where $\beta\leq\beta^{\prime}\leq\hat{\beta}$.*

This implies that there exist $A_{1},A_{2},\ldots,A_{s+\beta^{\prime}}$ in $\operatorname{\mathcal{F}}$ such that $\operatorname{\mathcal{F}}^{\prime}=\operatorname{\mathcal{F}}\setminus\{A_{1},A_{2},\ldots,A_{s+\beta^{\prime}}\}$ is an intersecting family. Without loss of generality assume that $\Delta(\operatorname{\mathcal{F}}^{\prime})$ has the maximum possible value. $|\operatorname{\mathcal{F}}^{\prime}|$ is equal to

[TABLE]

First we show that for each $i\leq s+\beta^{\prime}$, $1\not\in A_{i}$. If $\operatorname{\mathcal{F}}^{\prime}\subseteq\operatorname{\mathcal{S}}_{1}$, then by the minimality of $\ell(\operatorname{\mathcal{F}})$, each $A_{i}$ must be disjoint from at least one member of $\operatorname{\mathcal{F}}^{\prime}\subseteq\operatorname{\mathcal{S}}_{1}$, so $1\not\in\cup_{i=1}^{s+{\beta^{\prime}}}A_{i}$. If $\operatorname{\mathcal{F}}^{\prime}\not\subseteq\operatorname{\mathcal{S}}_{1}$, then by the Hilton-Milner theorem, we conclude that $|\operatorname{\mathcal{F}}^{\prime}|={n-1\choose k-1}-{n-k-1\choose k-1}+1$. Consequently, there exists a unique $B\in\operatorname{\mathcal{F}}^{\prime}$ such that $\operatorname{\mathcal{F}}^{\prime}\setminus\{B\}\subseteq\operatorname{\mathcal{S}}_{1}$ and moreover, we must have $t=2$, $\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor=k$ and $\beta^{\prime}=\hat{\beta}$. If there is $A_{i}$ such that $1\in A_{i}$, by the minimality of $\ell(\operatorname{\mathcal{F}})$, $A_{i}$ must be disjoint from $B$. Define $\operatorname{\mathcal{F}}^{\prime\prime}=(\operatorname{\mathcal{F}}^{\prime}\setminus\{B\})\cup\{A_{i}\}$. Hence, $|\operatorname{\mathcal{F}}^{\prime}|=|\operatorname{\mathcal{F}}^{\prime\prime}|$ and $\Delta(\operatorname{\mathcal{F}}^{\prime\prime})=\Delta(\operatorname{\mathcal{F}}^{\prime})+1$ which contradicts with the fact that $\Delta(\operatorname{\mathcal{F}}^{\prime})$ has the maximum possible value. Then, $1\not\in\cup_{i=1}^{s+{\beta^{\prime}}}A_{i}$. We now consider the following three subcases.

1. (a)

$\operatorname{\mathcal{F}}^{\prime}\subseteq\operatorname{\mathcal{S}}_{1}$* and $|T(A_{1},A_{2},\ldots,A_{s+\beta^{\prime}}:s)|=\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor$.*

Since $\beta\leq\beta^{\prime}\leq\hat{\beta}$, we have $\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor=\lfloor{(s+\beta)k\over\beta+1}\rfloor$. In view of the last part of the proof of Lemma 1 (b), there are $1\leq i_{1}<\ldots<i_{s}\leq s+\beta^{\prime}$ such that $T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)=A_{i_{1}}\cup\cdots\cup A_{i_{s}}.$ Also, note that for every $1\leq j_{1}<\ldots<j_{s}\leq s+\beta^{\prime}$, we have

[TABLE]

From this fact and since $\operatorname{\mathcal{F}}$ is an $(s,t)$-union intersecting family, the number of elements of $\operatorname{\mathcal{F}}^{\prime}$ which can be disjoint from $\cup_{\ell=1}^{s}A_{j_{\ell}}$ for some $s$ $k$-sets $A_{j_{1}},\ldots,A_{j_{s}}$ of $A_{i}$’s is at most $t-1$, say $F_{1},\ldots,F_{t-1}$. Therefore, $\operatorname{\mathcal{F}}^{\prime}\subseteq\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta^{\prime}}:s)\cup\{F_{1},\ldots,F_{t-1}\}$. Thus, by applying Lemma 1 (b), we obtain

[TABLE]

and consequently $\beta^{\prime}=\hat{\beta}$. Therefore,

[TABLE]

and equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to

[TABLE]

such that $|T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)|=\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor$, $F_{i}\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s+\beta^{\prime}}:s)$, and the family $\{A_{1},\ldots,A_{s+\beta^{\prime}},F_{1},\ldots,F_{t-1}\}$ is an $(s,t)$-union intersecting family. 2. (b)

$\operatorname{\mathcal{F}}^{\prime}\not\subseteq\operatorname{\mathcal{S}}_{1}$* and $|T(A_{1},A_{2},\ldots,A_{s+\beta^{\prime}}:s)|=\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor$.*

As $\operatorname{\mathcal{F}}^{\prime}\not\subseteq\operatorname{\mathcal{S}}_{1}$, there exist a $k$-set $B\in\operatorname{\mathcal{F}}^{\prime}$ such that $\operatorname{\mathcal{F}}^{\prime}\setminus\{B\}\subseteq\operatorname{\mathcal{S}}_{1}$ and we have $t=2$, $\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor=k$, and $\beta^{\prime}=\hat{\beta}$. Since $|T(A_{1},A_{2},\ldots,A_{s+\beta^{\prime}}:s)|=\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor=k$, in view of the last part of the proof of Lemma 1 (b), there are $1\leq i_{1}<\ldots<i_{s}\leq s+\beta^{\prime}$ such that $T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)=A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{s}}.$ As $|T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)|=k$, $s$ must be equal to $1$. Therefore $|T(A_{1},\ldots,A_{1+\beta^{\prime}}:1)|=k$. Since $T(A_{1},\ldots,A_{1+\beta^{\prime}}:1)=\cap_{i=1}^{1+\beta^{\prime}}A_{i}$, we obtain $\beta^{\prime}=0$. As $t=2$, $s=1$, and $\operatorname{\mathcal{F}}$ is $(s,t)$-union intersecting, there is a unique $B_{1}\in\operatorname{\mathcal{F}}^{\prime}$ that $A_{1}\cap B_{1}=\varnothing.$ One can check that $\operatorname{\mathcal{F}}^{\prime}\setminus\{B,B_{1}\}\subseteq\operatorname{\mathcal{S}}_{1}(A_{1},B:1)$. Therefore, $|\operatorname{\mathcal{F}}^{\prime}|\leq{n-1\choose k-1}-{n-k-1\choose k-1}-{n-k-2\choose k-2}+2$, a contradiction. 3. (c)

$|T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)|<\lfloor{(s+\beta)k\over\beta+1}\rfloor$.

There is at most one member $B\in\operatorname{\mathcal{F}}^{\prime}$ such that $\operatorname{\mathcal{F}}^{\prime}\setminus\{B\}\subseteq\operatorname{\mathcal{S}}_{1}.$ Since $\operatorname{\mathcal{F}}$ is an $(s,t)$-union intersecting family, every $s$ $k$-sets of $A_{i}$’s such as $A_{i_{1}},\ldots,A_{i_{s}}$ are disjoint from at most $t-1$ $k$-subsets in $\operatorname{\mathcal{F}}^{\prime}$. Therefore,

[TABLE]

Now by applying Lemma 1 (b), we obtain

[TABLE]

Since $|T(A_{1},\ldots,A_{s+\beta^{\prime}}:s)|<\lfloor{(s+\beta^{\prime})k\over\beta^{\prime}+1}\rfloor$ and $k\geq 3$, one can check that

[TABLE]

provided that $n$ is sufficiently large, which is not possible. 2. (ii)

$s+\hat{\beta}+1\leq\ell(\operatorname{\mathcal{F}})\leq M^{1-{1\over 3s}}.$

Let $\operatorname{\mathcal{F}}^{\prime}$ be a largest intersecting family of $\operatorname{\mathcal{F}}$. Hence, $|\operatorname{\mathcal{F}}^{\prime}|$ is at least

[TABLE]

As $M=O(n^{k-2})$, we have $M^{1-{1\over 3s}}=o(n^{k-2})$. Since $\lfloor{(s+\beta)k\over\beta+1}\rfloor\geq k$ and $M^{1-{1\over 3s}}=o(n^{k-2})$, if $n$ is sufficiently large, then we have

[TABLE]

By using Theorem B, $\operatorname{\mathcal{F}}^{\prime}$ is a star or a Hilton-Milner family. Therefore, without loss of generality we may assume that there exists at most one $B\in\operatorname{\mathcal{F}}^{\prime}$ such that $\operatorname{\mathcal{F}}^{\prime}\setminus\{B\}$ is a subfamily $\operatorname{\mathcal{S}}_{1}$.

First assume that $\lfloor{(s+\beta)k\over\beta+1}\rfloor\geq k+1$. By applying Lemma 1 (b) for $\operatorname{\mathcal{F}}^{\prime}\setminus\{B\}$ and one of $s+\hat{\beta}+1$ $k$-subsets of $\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{F}}^{\prime}$, we obtain

[TABLE]

Hence,

[TABLE]

Note that ${n-\lfloor{(s+\hat{\beta}+1)k\over\hat{\beta}+2}\rfloor-1\choose k-1}-{n-\lfloor{(s+\hat{\beta})k\over\hat{\beta}+1}\rfloor-1\choose k-1}={n-\lfloor{(s+\hat{\beta})k\over\hat{\beta}+1}\rfloor-1\choose k-2}$. Therefore, $|\operatorname{\mathcal{F}}|$ is at most

[TABLE]

This concludes that for sufficiently large $n$, $|\operatorname{\mathcal{F}}|$ is less than $M$, a contradiction.

Assume that $\lfloor{(s+\beta)k\over\beta+1}\rfloor=k$. Therefore, $\hat{\beta}=\beta$ and $\lfloor{(s+\beta+1)k\over\beta+2}\rfloor=k$. Take $A_{1},\ldots,A_{s+\beta+1}$ in $\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{F}}^{\prime}$. If we have $|T(A_{1},\ldots,A_{s+\beta+1}:s)|<k$, then by applying Lemma 1 (b), we obtain that there exists a positive constant $c$ such that

[TABLE]

This implies

[TABLE]

which is less than $M$ when $n$ is sufficiently large, a contradiction.

Assume that $|T(A_{1},\ldots,A_{s+\beta+1}:s)|=\lfloor{(s+\beta+1)k\over\beta+2}\rfloor=k$. In view of the last part of the proof of Lemma 1 (b), there are $1\leq i_{1}<\ldots<i_{s}\leq s+\beta+1$ such that

[TABLE]

This implies that $s$ must be equal to $1$. If $s=1$, then we have $T(A_{1},\ldots,A_{\beta+2}:1)=\cap_{i=1}^{\beta+2}A_{i}$ and hence $|T(A_{1},\ldots,A_{\beta+2}:1)|=|\cap_{i=1}^{\beta+2}A_{i}|\leq k-1$ which contradicts with $|T(A_{1},\ldots,A_{\beta+2}:1)|=k$. 3. (iii)

$\ell(\operatorname{\mathcal{F}})>M^{1-{1\over 3s}}$.

By Lemma B, we have $e(\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}])\geq{M^{2-{2\over 3s}}\over 2{2k\choose k}}$ and by Theorem G, $\operatorname{\mathcal{F}}$ contains a subgraph which is isomorphic to $K_{s,t}$ when $n$ is sufficiently large.

∎

Here we intend to elaborate on the $i$th largest $(s,t)$-union intersecting families for some $i$. Assume that $n$ is sufficiently large. Let $\{A_{1},\ldots,A_{s}\}$ be $s$ pairwise distinct $k$-subsets of $[n]$. By Definition 3 we know that $T(A_{1},\ldots,A_{s}:s)=\cup_{i=1}^{s}A_{i}$. Define

[TABLE]

where $F_{i}\in\operatorname{\mathcal{S}}_{1}\setminus\operatorname{\mathcal{S}}_{1}(A_{1},\ldots,A_{s}:s).$ By using Inequality (2), one can verify that $|\operatorname{\mathcal{L}}|$ is equal to

[TABLE]

Let $n=n(k,s)$ be sufficiently large and $s\geq 2$. If $\lfloor{(s+1)k\over 2}\rfloor<|T(A_{1},\ldots,A_{s}:s)|\leq sk$, then by using Theorem 1, $\operatorname{\mathcal{L}}$ is the $i$th largest $(s,t)$-union intersecting family, where $i=sk-|T(A_{1},\ldots,A_{s}:s)|+2$.

If $|T(A_{1},\ldots,A_{s}:s)|=\lfloor{(s+1)k\over 2}\rfloor$, then $|\operatorname{\mathcal{L}}|$ is equal to ${n-1\choose k-1}-{n-\lfloor{(s+1)k\over 2}\rfloor-1\choose k-1}+s+t-1$. Let $\{A^{\prime}_{1},\ldots,A^{\prime}_{s},A^{\prime}_{s+1}\}$ be $s+1$ pairwise distinct $k$-subsets of $[n]$ such that $T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)=\lfloor{(s+1)k\over 2}\rfloor$. Define

[TABLE]

We have $|\operatorname{\mathcal{L}}^{\prime}|$ is equal to ${n-1\choose k-1}-{n-\lfloor{(s+1)k\over 2}\rfloor-1\choose k-1}+s+t$ which is greater than $|\operatorname{\mathcal{L}}|$. Therefore, $\operatorname{\mathcal{L}}^{\prime}$ and $\operatorname{\mathcal{L}}$ are the $(\lfloor{(s-1)k\over 2}\rfloor+2)$th and $(\lfloor{(s-1)k\over 2}\rfloor+3)$th largest $(s,t)$-union intersecting families, respectively.

Now assume that there are $\{A_{1},\ldots,A_{s}\}$ and $\{A^{\prime}_{1},\ldots,A^{\prime}_{s+1}\}$ such that $|T(A_{1},\ldots,A_{s}:s)|=\lfloor{(s+1)k\over 2}\rfloor-1$ and $|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|=\lfloor{(s+1)k\over 2}\rfloor-1$. If $(s+1)k$ is even, then $2|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|=(s+1)k-2$. Therefore, there are at most two members  in $\cup_{i=1}^{s+1}A^{\prime}_{i}$ such that each of them appears in one of $A^{\prime}_{i}$’s. If for each $i\leq s+1$ we have $A^{\prime}_{i}\subset T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)$, then by using Inequality (2), we have

[TABLE]

If for only one $i\leq s+1$ we have $A^{\prime}_{i}\not\subseteq T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)$, then one can construct an $(s,t)$-union intersecting family $\operatorname{\mathcal{L}}^{\prime}_{1}$ with $\ell(\operatorname{\mathcal{L}}^{\prime}_{1})=s+1$ and

[TABLE]

Now suppose that $A^{\prime}_{i}\not\subseteq T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)$ and $A^{\prime}_{j}\not\subseteq T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)$ for exactly two $1\leq i\neq j\leq s+1$. By using Inequality 2, the number of $A\in\operatorname{\mathcal{S}}_{1}$ which has no common element with $T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)$ and intersects at least two of $A^{\prime}_{i}$’s is ${n-|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|-3\choose k-3}$. Therefore, for $0\leq m\leq t-1$, one can construct a maximal $(s,t)$-union family $\operatorname{\mathcal{L}}^{\prime}_{2,m}$ with $\ell(\operatorname{\mathcal{L}}^{\prime}_{2,m})=s+1$ and

[TABLE]

Therefore, we have some different types $(s,t)$-union intersecting families with $\ell(\operatorname{\mathcal{F}})=s+1$, $|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|=\lfloor{(s+1)k\over 2}\rfloor-1$, and different sizes and one type of $(s,t)$-union intersecting families with $\ell(\operatorname{\mathcal{F}})=s$, $|T(A_{1},\ldots,A_{s}:s)|=\lfloor{(s+1)k\over 2}\rfloor-1$.

If $(s+1)k$ is odd, then $2|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|=(s+1)k-3$. Therefore, there are at most three members in $\cup_{i=1}^{s+1}A^{\prime}_{i}$ such that each of them appears in one of $A^{\prime}_{i}$’s. Using the same discussion as above one can find some different types of $(s,t)$-union intersecting families with $\ell(\operatorname{\mathcal{F}})=s+1$, $|T(A^{\prime}_{1},\ldots,A^{\prime}_{s+1}:s)|=\lfloor{(s+1)k\over 2}\rfloor-1$ and different sizes.

In the proof of Theorem 2, we need the following theorem by Frankl [13] and independently Kalai [26] which is a generalization of a classical result due to Bollobás [3].

Theorem H**.**

[13, 26]* Let $\{(A_{1},B_{1}),\ldots,(A_{h},B_{h})\}$ be a family of pairs of subsets of an arbitrary set with $|A_{i}|=k$ and $|B_{i}|=\ell$ for all $1\leq i\leq h$. If $A_{i}\cap B_{i}=\varnothing$ for $1\leq i\leq h$ and $A_{i}\cap B_{j}\neq\varnothing$ for $1\leq i<j\leq h$, then $h\leq{k+\ell\choose k}$.*

For simplicity of notation, for each $1\leq i\leq k-1$, define N_{i}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }{n-1\choose k-1}-{n-k\choose k-1}+{n-k-i\choose k-i-1} and for $k$ define N_{k}\mbox{\ \stackrel{{\scriptstyle\rm def}}{{=}}\ }{n-1\choose k-1}-{n-k\choose k-1}. Note that for $1\leq i\leq k-1$, we have $N_{i-1}-N_{i}={n-k-i\choose k-i}=\Omega(n^{k-i}).$

Proof of Theorem 2.

First we show that $\ell(\operatorname{\mathcal{F}})\leq{2k-1\choose k-1}(t-1)$. If $t=1$, $\operatorname{\mathcal{F}}$ is intersecting and hence $\ell(\operatorname{\mathcal{F}})=0$. Assume that $t\geq 2$ and $\operatorname{\mathcal{F}}$ is not intersecting. Therefore, there exists some disjoint pair in $\operatorname{\mathcal{F}}$. For a $k$-set $A$, define $N(A)=\{B\in{[n]\choose k}|\,A\cap B=\varnothing\}$. Define $\operatorname{\mathcal{F}}_{1}=\operatorname{\mathcal{F}}$. For each $i\geq 2$, if there exists some disjoint pair in $\operatorname{\mathcal{F}}_{i-1}$, choose $B_{i-1}\in\operatorname{\mathcal{F}}_{i-1}$ and $C_{i-1}\in N(B_{i-1})\cap\operatorname{\mathcal{F}}_{i-1}$ and define $\operatorname{\mathcal{F}}_{i}=\operatorname{\mathcal{F}}_{i-1}\setminus(N(B_{i-1})$. Let $m$ be the largest index $i$ for which $\operatorname{\mathcal{F}}_{i}$ contains some disjoint pair. For $m+1\leq j\leq 2m$, set $B_{j}=C_{2m-j+1}$ and $C_{j}=B_{2m-j+1}$. One may check that the family$\{(B_{1},C_{1}),\ldots,(B_{2m,}C_{2m})\}$ satisfies the condition of Theorem H for $l=k$ and consequently $m\leq{2k-1\choose k-1}$. Let $\operatorname{\mathcal{N}}$ be a subfamily of $\operatorname{\mathcal{F}}$ defined as follows

[TABLE]

Since $\operatorname{\mathcal{F}}$ is $(1,t)$-union intersecting, one can verify that $|\operatorname{\mathcal{N}}|\leq m(t-1)$. Note that $\operatorname{\mathcal{F}}_{m+1}$ is an intersecting family and $\operatorname{\mathcal{F}}$ is disjoint union of $\operatorname{\mathcal{F}}_{m+1}$ and $\operatorname{\mathcal{N}}$. This yields $\ell(\operatorname{\mathcal{F}})\leq|\operatorname{\mathcal{N}}|\leq{2k-1\choose k-1}(t-1)$.

Assume that $|\operatorname{\mathcal{F}}|=N_{\gamma}+\gamma t$. Let $\operatorname{\mathcal{F}}^{*}$ be one of largest intersecting subfamilies of $\operatorname{\mathcal{F}}$ such that $\Delta(\operatorname{\mathcal{F}}^{*})$ has the maximum possible value. Assume that $\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{F}}^{*}=\{A_{1},\ldots,A_{\ell(\operatorname{\mathcal{F}})}\}$. Therefore, $|\operatorname{\mathcal{F}}^{*}|=|\operatorname{\mathcal{F}}|-\ell(\operatorname{\mathcal{F}}).$ Consider the following three cases.

1. (1)

$\ell(\operatorname{\mathcal{F}})=\gamma$* and $\operatorname{\mathcal{F}}^{*}\subseteq\operatorname{\mathcal{S}}_{1}$. *

We have $|\operatorname{\mathcal{F}}^{*}|=N_{\gamma}+\gamma(t-1)$. Since $\ell(\operatorname{\mathcal{F}})=\gamma$ and $\operatorname{\mathcal{F}}=\operatorname{\mathcal{F}}^{*}\cup\{A_{1},\ldots A_{\gamma}\}$, each $A_{j}$ is disjoint from at least one member of $\operatorname{\mathcal{F}}^{*}$ and hence $1\not\in\cup_{j=1}^{\gamma}A_{j}$. Then

[TABLE]

Since $\gamma\leq k-2$, by applying Lemma 1 (c), we conclude that $|\operatorname{\mathcal{F}}^{*}\setminus(\cup_{j=1}^{\gamma}N(A_{j}))|\leq N_{\gamma}$. Since $\operatorname{\mathcal{F}}$ is $(1,t)$-union intersecting, for each $j$, $A_{j}$ is disjoint from at most $t-1$ members of $\operatorname{\mathcal{F}}$. As for each $j$, $|N(A_{j})\cap\operatorname{\mathcal{F}}|\leq t-1$, $|\operatorname{\mathcal{F}}|=N_{\gamma}+\gamma t$, and

[TABLE]

we have $\operatorname{\mathcal{F}}$ is a disjoint union of

[TABLE]

Moroever, for each $j$, we have $|N(A_{j})\cap\operatorname{\mathcal{F}}|=t-1$, $N(A_{j})\cap\operatorname{\mathcal{F}}\subseteq\operatorname{\mathcal{F}}^{*}\subseteq\operatorname{\mathcal{S}}_{1}$, and $|\operatorname{\mathcal{F}}^{*}\setminus(\cup_{j=1}^{\gamma}N(A_{j}))|=N_{\gamma}$. From the last equality and by using Lemma 1 (c), we obtain

[TABLE]

and $|\cap_{j=1}^{\gamma}A_{j}|=k-1.$ By taking $E=\cap_{j=1}^{\gamma}A_{j}$ and $J=\{1\}\cup(\cup_{j=1}^{\gamma}A_{j}\setminus E)$ in Definition 1, one can see that $\operatorname{\mathcal{F}}\setminus(\cup_{j=1}^{\gamma}N(A_{j}))$ is isomorphic to $\operatorname{\mathcal{J}}_{\gamma}$. For each $j\leq\beta+1$, by taking $\operatorname{\mathcal{B}}_{j}=N(A_{j})\cap\operatorname{\mathcal{F}}$ in Definition 5, one can check that $\operatorname{\mathcal{F}}$ is isomorphic to $\operatorname{\mathcal{J}}^{1,t}_{\gamma}$. By Theorem C, $\operatorname{\mathcal{F}}^{*}$ is either a star or isomorphic to a subfamily $\operatorname{\mathcal{J}}_{i}$ where $0\leq i\leq\gamma-1$. First let $\operatorname{\mathcal{F}}^{*}$ be a star and $\operatorname{\mathcal{F}}^{*}\subseteq\operatorname{\mathcal{S}}_{1}$. 2. (2)

$\gamma+1\leq\ell(\operatorname{\mathcal{F}})\leq{2k-1\choose k-1}(t-1)$* and $\operatorname{\mathcal{F}}^{*}\subseteq\operatorname{\mathcal{S}}_{1}$. *

Let $A_{1},\ldots,A_{\gamma+1}\in\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{F}}^{*}.$ By using minimality of $\ell(\operatorname{\mathcal{F}})$, each $A_{i}$ is disjoint from at least one member of $\operatorname{\mathcal{F}}^{*}$. Therefore, $1\not\in A_{i}$ for each $i\leq\gamma+1.$ Then

[TABLE]

and by applying Lemma 1 (c), we obtain $|\operatorname{\mathcal{F}}^{*}\setminus((\cup_{i=1}^{\gamma+1}N(A_{i}))|\leq N_{\gamma+1}$. Since

[TABLE]

we have $|\operatorname{\mathcal{F}}|\leq N_{\gamma+1}+(\gamma+1)(t-1)+\ell(\operatorname{\mathcal{F}})<N_{\gamma}$, which is not possible when $n$ is sufficiently large. 3. (3)

$\gamma\leq\ell(\operatorname{\mathcal{F}})\leq{2k-1\choose k-1}(t-1)$* and $\operatorname{\mathcal{F}}^{*}$ is not a star. *

By Theorem C, $\operatorname{\mathcal{F}}^{*}\subseteq\operatorname{\mathcal{J}}_{c}$ for some $1\leq c\leq\beta+1$. Then, for some $b\leq c$, there exist $B_{1},\ldots,B_{b}\in\operatorname{\mathcal{F}}^{*}$ such that $\operatorname{\mathcal{F}}^{*}\setminus\{B_{1},\ldots,B_{b}\}\subseteq\operatorname{\mathcal{S}}_{1}$ and $B_{j}\not\in\operatorname{\mathcal{S}}_{1}$. At most $b-1$ of $A_{1},\ldots,A_{\gamma}$ contain $1$; otherwise if for $1\leq j_{1}\leq\cdots\leq j_{b}\leq\gamma$ we have $1\in\cap_{i=1}^{b}A_{j_{i}}$, then $\operatorname{\mathcal{F}}^{\prime}=(\operatorname{\mathcal{F}}^{*}\setminus\{B_{1},\ldots,B_{b}\})\cup\{A_{j_{1}},\ldots,A_{j_{b}}\}$ is an intersecting family with $|\operatorname{\mathcal{F}}^{\prime}|=|\operatorname{\mathcal{F}}^{*}|$ and $\Delta(\operatorname{\mathcal{F}}^{\prime})>\Delta(\operatorname{\mathcal{F}}^{*})$, which contradicts with the fact that $\Delta(\operatorname{\mathcal{F}}^{*})$ has the maximum possible value. Therefore, without loss of generality we can assume that $A_{1},\ldots,A_{b^{\prime}}$ do not contain $1$ for $b^{\prime}=\gamma+1-b$. Hence,

[TABLE]

and by Lemma 1 (c), we obtain $|\operatorname{\mathcal{F}}^{*}\setminus((\cup_{j=1}^{b^{\prime}}N(A_{j}))\cup\{B_{1},\ldots,B_{b}\})|\leq N_{\gamma+1}$. Since

[TABLE]

we obtain $|\operatorname{\mathcal{F}}|\leq N_{\gamma+1}+b+b^{\prime}(t-1)+\ell(\operatorname{\mathcal{F}})<N_{\gamma}$, which is not possible when $n$ is sufficiently large.

∎

For the proof of Theorem 3 we need to use the well-known Erdős-Stone-Simonovits theorem [10, 11]. For a given graph $G$, the Turán number ${\rm ex}(n,G)$ is defined to be the maximum number of edges in a graph with $n$ vertices containing no subgraph isomorphic to $G$. The Erdős-Stone-Simonovits theorem asserts that for any graph $G$ with $\chi(G)\geq 2$, ${\rm ex}(G,n)=(1-{1\over\chi(G)-1}){n\choose 2}+o(n^{2}).$

Proof of Theorem 3.

The proof is by induction on $r$. By Theorem 1, the assertion is true when $r=1$. Let $r\geq 2$. Suppose now that the assertion is true for $r-1$. Also, without loss of generality suppose that

[TABLE]

Consider the following cases.

1. (1)

$\max_{i\in[n]}|\operatorname{\mathcal{F}}\cap\operatorname{\mathcal{S}}_{i}|\leq{n-1\choose k-1}-{n-\sum_{j=2}^{r+1}s_{j}k-1\choose k-1}+s_{1}$.

Then the number of disjoint pair in $\operatorname{\mathcal{F}}$ is at least

[TABLE]

provided that $n$ is large enough. Hence, by the Erdős-Stone-Simonovits theorem $\operatorname{\operatorname{KG}}_{n,k}[\operatorname{\mathcal{F}}]$ contains some subgraph isomorphic to $K_{s_{1},s_{2},\ldots,s_{r+1}}$ provided that n is large enough, which is a contradiction. 2. (2)

$\max_{i\in[n]}|\operatorname{\mathcal{F}}\cap\operatorname{\mathcal{S}}_{i}|>{n-1\choose k-1}-{n-\sum_{j=2}^{r+1}s_{j}k-1\choose k-1}+s_{1}$.

Without loss of generality assume that $\max_{i\in[n]}|\operatorname{\mathcal{F}}\cap\operatorname{\mathcal{S}}_{i}|=|\operatorname{\mathcal{F}}\cap\operatorname{\mathcal{S}}_{n}|$. Notice that one can assume that $\operatorname{\mathcal{S}}_{n}\subset\operatorname{\mathcal{F}}$. Because otherwise, if $\operatorname{\mathcal{S}}_{n}\not\subset\operatorname{\mathcal{F}}$, then $|\operatorname{\mathcal{F}}\cap\operatorname{\mathcal{S}}_{n}|<{n-1\choose k-1}$. Therefore,

[TABLE]

By induction hypothesis $\operatorname{\operatorname{KG}}_{n-1,k}[\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{S}}_{n}]$ contains $K_{s_{2},\ldots,s_{r+1}}$. As

[TABLE]

one can greedily pick $s_{1}$ sets of $\operatorname{\mathcal{S}}_{n}$ such that constructs a copy of $K_{s_{1},s_{2},\ldots,s_{r+1}}$ in $\operatorname{\operatorname{KG}}_{n,k}[F]$, a contradiction. Since $\operatorname{\mathcal{S}}_{n}$ is a subfamily of $\operatorname{\mathcal{F}}$, same as previous discussion $K_{s_{2},\ldots,s_{r+1}}$ cannot be contained in $\operatorname{\operatorname{KG}}_{n-1,k}[\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{S}}_{n}]$. Therefore, by induction hypothesis, we have

[TABLE]

and the equality holds if and only if $\operatorname{\mathcal{F}}\setminus\operatorname{\mathcal{S}}_{n}$ is isomorphic to

[TABLE]

such that $|T(A_{1},A_{2},\ldots,A_{s_{r+1}+\hat{\beta}})|=\lfloor{(s_{r+1}+\beta)k\over\beta+1}\rfloor$, $F_{i}\in\operatorname{\mathcal{S}}_{r-1}\setminus\operatorname{\mathcal{S}}_{r-1}^{[r-2]}(A_{1},A_{2},\ldots,A_{s+\hat{\beta}}:s)$, and $F_{i}\cap[r-2]=\varnothing$ for each $i$ ( Note that in this step all families are subfamilies of ${[n-1]\choose k}$ because we remove $\operatorname{\mathcal{S}}_{n}$ from $\operatorname{\mathcal{F}}$ so we do not meet $n$.).

Thus,

[TABLE]

and the equality holds if and only if $\operatorname{\mathcal{F}}$ is isomorphic to

[TABLE]

such that $|T(A_{1},A_{2},\ldots,A_{s_{r+1}+\hat{\beta}})|=\lfloor{(s_{r+1}+\beta)k\over\beta+1}\rfloor$, $F_{i}\in\operatorname{\mathcal{S}}_{r}\setminus\operatorname{\mathcal{S}}_{r}^{[r-1]}(A_{1},A_{2},\ldots,A_{s+\hat{\beta}}:s)$, and $F_{i}\cap[r-1]=\varnothing$ for each $i$.

∎

Proofs of Theorem 4 and Corollary 3 are the same as the proof of Theorem 3.

Acknowledgements

The author is grateful to Meysam Alishahi and Amir Daneshgar for their valuable comments. This research was in part supported by a grant from IPM (No. 98050012).