Algebraic approach to stability results for Erdős-Ko-Rado theorem

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ is intersecting if $F_{1}\cap F_{2}\neq\emptyset$ for every pair of sets $F_{1},F_{2}\in\mathcal{F}$. A fundamental result in extremal set theory, due to Erdős, Ko, and Rado [7], determines the maximum size of a $k$-uniform intersecting family $\mathcal{F}$.

Note that the only extremal family in the Erdős-Ko-Rado theorem is the star, also known as a trivially intersecting family. Hilton and Milner [21] established a stability result, showing that the size of non-trivially intersecting families is significantly smaller than the maximum given by the Erdős-Ko-Rado theorem. Before presenting a series of stability results, we first introduce some necessary notations. For a set system $\mathcal{F}\subseteq 2^{[n]}$, we define the maximum degree of $\mathcal{F}$ as

$$d_{\textup{max}}(\mathcal{F}):=\max\limits_{i\in[n]}|\{F\in\mathcal{F}:i\in F\}|.$$

We use $[i,j]$ to denote the set $\{i,i+1,\ldots,j\}$. Let $k\geq 3$ be a positive integer, we will use $\binom{[n]}{k}$ to denote the family of all subsets of $[n]$ with size $k$. For $i\in[3,k+1]$, we define

$$\mathcal{M}_{i}:=\bigg{\{}F\in\binom{[n]}{k}:1\in F,F\cap[2,i]\neq\emptyset\bigg{\}}\cup\bigg{\{}F\in\binom{[n]}{k}:1\notin F,[2,i]\subseteq F\bigg{\}}.$$

It is not hard to check that $\mathcal{M}_{i}$ is intersecting and $|\mathcal{M}_{i}|=\sum\limits_{j=2}^{i}\binom{n-j}{k-2}+\binom{n-i}{k-i+1}$. In particular, we can see that $|\mathcal{M}_{3}|=\binom{n-2}{k-2}+2\binom{n-3}{k-2}=\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}+\binom{n-4}{k-3}=|\mathcal{M}_{4}|$. Moreover, we can see $d_{\textup{max}}(\mathcal{M}_{i})=\sum\limits_{j=2}^{i}\binom{n-j}{k-2}$. For $j\in[1,n-k]$ we further define

	$\displaystyle\mathcal{M}_{k,j}:=\bigg{\{}F\in\binom{[n]}{k}:1\in F,F\cap[2,k]\neq\emptyset\bigg{\}}\cup\bigg{\{}F\in\binom{[n]}{k}:1\in F,F\cap[2,k]=\emptyset,[k+1,k+j]\subseteq F\bigg{\}}$
	$\displaystyle\cup\bigg{\{}F\in\binom{[n]}{k}:1\notin F,[2,k]\subseteq F,|F\cap[k+1,k+j]|=1\bigg{\}}.$

In particular, one can easily check that $\mathcal{M}_{k,n-k}=\mathcal{M}_{k},\mathcal{M}_{k,1}=\mathcal{M}_{k+1}$.

For two distinct sets $E_{1},E_{2}\in\binom{[n]}{k}$ with $\left|E_{1}\cap E_{2}\right|=k-2$, and an element $x_{0}\in[n]\setminus\left(E_{1}\cup E_{2}\right)$, we define

$$\mathcal{K}\left(E_{1},E_{2},x_{0}\right):=\left\{G\in\binom{[n]}{k}:x_{0}\in G,G\cap E_{1}\neq\emptyset,G\cap E_{2}\neq\emptyset\right\}\cup\left\{E_{1},E_{2}\right\},$$

and we write $\mathcal{K}_{2}$ for any family isomorphic to $\mathcal{K}\left(E_{1},E_{2},x_{0}\right)$.

If the intersecting family $\mathcal{F}\subseteq\binom{[n]}{k}$ is neither extremal Erdős-Ko-Rado family nor extremal Hilton-Milner family, what is the maximum size of $\mathcal{F}$? Han and Kohayakawa [20] resolved this problem. Here we use $\mathcal{F}_{\textup{EKR}}$ and $\mathcal{F}_{\textup{HM}}$ to denote the corresponding extremal families in Theorems 1.1 and 1.2 respectively.

Han and Kohayakawa [20] further asked what the next maximum intersecting $k$-uniform families on $[n]$ are? Kostochka and Mubayi [25] answered this question when $n$ is sufficiently large. Recently, Huang and Peng [22] completely answered this question for any $n\geq 2k+1$ as follows.

The theorems above can be viewed as the 1st, 2nd, and 3rd-level full stability results for the Erdős-Ko-Rado theorem, respectively. Currently, there are several different proofs for these stability results, particularly regarding the Hilton-Milner theorem [9, 11, 15, 23, 28, 29] and some interesting variants [12, 27]. However, as far as we know, no work has emerged that approaches proving these stability results from multilinear polynomial methods. We aim to fill this gap.

The main contribution of this paper is to provide an algebraic approach to proving the stability result at the $t$-th level of the Erdős-Ko-Rado theorem for arbitrary positive integer $t$. To more clearly demonstrate our method, we will present a slightly weaker version of the stability result under the following conditions. Specifically, we make two assumptions: first, for the stability result at the $t$-th level, we assume $k\geq t+2$; second, we assume $n>\frac{(5+\sqrt{5})k-7}{2}\approx 3.618k$ instead of $n>2k$.

Note that with our framework, combined with some simple and appropriate structural analysis, we can also obtain stability results in Theorems 1.2, 1.3 and 1.4 mentioned earlier, as well as higher-layer stability results. However, since the main purpose of this paper is to present a new framework and method, we will not fully expand on these full proofs. For interested readers, we will provide an alternative proof of Theorem 1.3 in the Appendix using our unified framework.

We remark that in Theorem 1.5, we provide the upper bound for $|\mathcal{F}|$ under the assumption that we prohibit all of the extremal structures up to the first $t-1$ levels. Moreover, we can characterize the corresponding extremal structures. After completing this draft, we are informed by Jian Wang that recent work by Kupavskii, and by Frankl and Wang [14, 16, 26] established a similar upper bound under the condition that the diversity $\gamma(\mathcal{F})=|\mathcal{F}|-d_{\textup{max}}(\mathcal{F})$ is large. Although we do not use the concept of diversity here, it has proven valuable in the study of $k$-uniform intersecting families. We recommend the interested readers to [10, 13, 14, 26, 27] and the references therein. Instead, we will introduce our framework and give the self-contained and elementary proofs in Section 3.2. When we finally try to analyze the extremal structures in Section 3.3 and Appendix, for convenience, we will use a celebrated result of Frankl [8], see Theorem 2.3.

The following triangular criterion is useful when we want to prove a sequence of polynomials to be linearly independent.

We follow the notations on a proof of Erdős-Ko-Rado theorem via multilinear polynomials [18]. Suppose that $\mathcal{F}:=\{F_{1},F_{2},\ldots,F_{m}\}$, where $F_{i}\subseteq[n]$ for each $1\leq i\leq m$. For a set $P\subseteq[n]$ and a non-negative integer $\beta$, we say the set $F$ satisfies the property $(P,\beta)$-intersection if $|F\cap P|=\beta$. Now suppose for each $F_{i}\in\mathcal{F}$, we write a collection of $s$ intersection properties (allow repetition) as

$$R_{i}=\{(P_{i_{1}},\beta_{i_{1}}),(P_{i_{2}},\beta_{i_{2}})),\ldots,(P_{i_{s}},\beta_{i_{s}})\}.$$

In [18], the authors built the relation between the multilinear polynomials and the certain collections of intersection properties, here we introduce the following key lemma and the proof in details.

When we analyze the structural properties of set systems in proofs of Theorem 1.3 and Theorem 1.5, we will take advantage of the following result of Frankl [8].

Our main approach is based on a robust linear algebra method developed in recent work of Gao, Liu and the second author [19]. Here we first show an example and explain how the standard linear algebra method works and then summarize some interesting tricks. Furthermore, we will briefly introduce the main ideas on the robust linear algebra method. For more on the linear algebra methods in combinatorics, we recommend the interested readers to the great textbook [3] and a recent note [32].

Suppose that $\mathcal{F}\subseteq 2^{[n]}$ is an $L$-intersecting family for some subset $L\subseteq[n]$ with $|L|=s$, Frankl and Wilson [17] showed that $|\mathcal{F}|\leq\sum\limits_{i=0}^{s}\binom{n}{i}$. We sketch the shorter proof of the above result by Babai [2] as follows. Let $\mathcal{F}=\{F_{1},F_{2},\ldots,F_{m}\}$, indexed so that $|F_{1}|\leq\cdots\leq|F_{m}|$ and $L=\{\ell_{1},\ldots,\ell_{s}\}$. For each $i$ let $\vec{v}_{i}$ be the incidence vector of $F_{i}$. Define polynomials $f_{1},f_{2},\ldots,f_{m}$ by $f_{i}(\vec{x})=\prod\limits_{\ell_{k}<|F_{i}|}(\vec{x}\cdot\vec{v}_{i}-\ell_{k})$. Then one can easily prove that $\{f_{i}\}_{i=1}^{m}$ are linearly independent via Proposition 2.1. Thus $|\mathcal{F}|$ can be upper bounded by the number of all possible polynomials. The first trick one can use is multilinear reduction, that is, we can replace each $x_{i}^{t}$ term by $x_{i}$ for any $1\leq i\leq n$ and $t\geq 2$ because $x_{i}\in\{0,1\}$. Using this trick, one can efficiently give the upper bound $\sum\limits_{i=0}^{s}\binom{n}{i}$ for $|\mathcal{F}|$ as the total degree of each monomial is at most $s$.

The second trick is that, one can further add more associated polynomials. For example, we can add $\sum_{i=0}^{s-1}\binom{n-1}{i}$ many extra polynomials in the following way. Label the sets in $\binom{[n-1]}{\leq s}$ with label $B_{i}$ for $1\leq i\leq q=\sum_{i=0}^{s-1}\binom{n-1}{i}$ such that $|B_{i}|\leq|B_{j}|$ when $i<j$. Let $\vec{w}_{i}$ be the characteristic vector of $B_{i}$, and let $h_{B_{i}}(\vec{x})=\prod_{j\in B_{i}}x_{j}$ for $i>1$. Then define a multilinear polynomial $g_{B_{i}}$ in $n$ variables as follows. $g_{B_{1}}=x_{n}-1$ and $g_{B_{i}}=(x_{n}-1)h_{B_{i}}(\vec{x})$ for $i>1$. In [33], Snevily proved that $\{f_{i}\}_{i=1}^{m}$ and $\{g_{B_{j}}\}_{j=1}^{q}$ are linearly independent. Thus the upper bound for $|\mathcal{F}|$ can be improved to $\sum_{i=0}^{s}\binom{n}{i}-\sum_{i=0}^{s-1}\binom{n-1}{i}=\sum_{i=0}^{s}\binom{n-1}{i}$. This trick has been applied to several problems, for example, see [1, 4, 5, 6, 24, 30, 31]. In a word, this trick consists of two parts, the first step is to choose some appropriate extra polynomials, one can see that in many previous works involved with this trick, the extra polynomials usually are clear and natural, which associate some explicit family of subsets. For instance in the above famous case, the extra polynomials associate to $\binom{[n-1]}{\leq s}$ exactly. The second part is that we need to show the union of the original polynomials and the extra polynomials are linearly independent. Usually the second part is much more complicated and difficult. Once we prove the linear independence, we can immediately see the improvement on the bounds for size of the family.

In the proofs of stability results of Kleitman’s isodiametric inequality, the authors in [19] mainly focus on another direction about the second trick. More precisely, one can first carefully choose a family of subsets which satisfies some appropriate properties and then associate each subset of the chosen family with the extra polynomial one by one. Then it will be easier to prove the linear independence. At the cost, one cannot know all of information about the chosen family, but sometimes one can ignore it, e.g., see the proof in [19, Theorem 1.10]. While in some cases, then the main task in the robust linear algebra method is to dig out the structural properties of the family one chooses, to achieve this, usually one can apply some structural analysis, e.g., see the proof in [19, Theorem 1.8].

There are several advantages in this robust linear algebra method. The first is that the linear independence usually will be easier to show. The second is that, usually the previous linear algebra methods just provide the bound of the size, while the new method can be used to prove some stability results, that means, we can not only capture the size of the family, but also obtain some structural information. We believe that this method has the potential to be applied to a wider range of problems.

Although our proofs are relatively simple, it might be helpful to briefly outline the main ideas.

1. 1.

   Following the ideas in [18], we partition the family $\mathcal{F}$ into two sub-families, $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$, where the sets in $\mathcal{F}_{1}$ contain the element that attains the maximum degree in $\mathcal{F}$. We then introduce two auxiliary families, $\mathcal{H}$ and $\mathcal{G}$. At this stage, we can associate these families with certain intersection conditions, which leads to an algebraic proof of [18].

2. 2.

   The key ingredient involves finding one more auxiliary family $\mathcal{S}$, which is a sub-family of the non-shadows of $\mathcal{F}_{1}$. We carefully associate $\mathcal{S}$ with appropriate intersection conditions. The first crucial point occurs at 3.1, after proving that the families $\mathcal{F}_{1}$, $\mathcal{G}$, $\mathcal{H}$, and $\mathcal{S}$ satisfy certain properties analogous to linear independence in normal linear algebra method, we obtain an important quantitative relation: $|\mathcal{F}_{1}|\leq\binom{n-1}{k-1}-|\mathcal{S}|$, leading to the inequality $|\mathcal{F}|\leq\binom{n-1}{k-1}-|\mathcal{S}|+|\mathcal{F}_{0}|$.

3. 3.

   To show the upper bound on $|\mathcal{F}|$, it is sufficient to analyze the lower bound on $|\mathcal{S}|-|\mathcal{F}_{0}|$. We establish a general lower bound on $|\mathcal{S}|$ in 3.3, and later, through a stability argument, refine this bound in 3.4, which is essential for determining the extremal structures.

4. 4.

   The final task is to carefully compare the specific sizes of several numbers. To do this, we establish inequalities between variables, explore monotonicity, and analyze the structure at extreme values in 3.5 and 3.6. Additionally, we will use Theorem 2.3 of Frankl. These steps are relatively straightforward and follow from natural analytical considerations.

For positive integers $n\geq 2k+1$, without loss of generality, we can assume that $\mathcal{F}$ is a maximal non-trivial intersecting family of $\binom{[n]}{k}$, that is, if we add any new member of $\binom{[n]}{k}$ to $\mathcal{F}$, then $\mathcal{F}$ is not non-trivial intersecting. For a family $\mathcal{F}\in\binom{[n]}{k}$, we denote the shadow of $\mathcal{F}$ as $\partial_{k-1}{\mathcal{F}}:=\{T\in\binom{[n]}{k-1}:T\subseteq F\ \text{for\ some\ }F\in\mathcal{F}\}$. Let $p\in[n]$ be the element which attains the maximum degree in $\mathcal{F}$. Consider the following families:

• •

  $\mathcal{F}_{0}:=\{F\in\mathcal{F}:p\notin F\}$;

• •

  $\mathcal{H}:=\{H\subseteq[n]:p\notin H,0\leq|H|\leq k-2\}$;

• •

  $\mathcal{F}_{1}:=\{F\in\mathcal{F}:p\in F\}$;

• •

  $\mathcal{G}:=\{G\subseteq[n]:p\in G,1\leq|G|\leq k-1\}$;

• •

  $\mathcal{S}:=\{S\subseteq\binom{[n]}{k-1}\setminus\partial_{k-1}{\mathcal{F}_{1}}:p\notin S,\exists F\in\mathcal{F}_{0}\textup{ such\ that\ }S\cap F=\emptyset\}$.

Let $\mathcal{A}:=\mathcal{F}_{1}\sqcup\mathcal{H}\sqcup\mathcal{G}\sqcup\mathcal{S}=\{A_{1},\ldots,A_{m}\}$. We define an ordering $\prec$ on the sets, and for two families $\mathcal{A},\mathcal{B}$, denote $\mathcal{A}\prec\mathcal{B}$ if and only if for any $A\in\mathcal{A}$ and $B\in\mathcal{B}$, we have $A\prec B$. We first arrange the sets in a linear order as follows: $\mathcal{H}\prec\mathcal{F}_{1}\prec\mathcal{G}\prec\mathcal{S}$. We put the members of $\mathcal{F}_{1}$ and $\mathcal{S}$ in arbitrary order and the members of $\mathcal{H}$ and $\mathcal{G}$ in order increasing by size, for example, $H_{i}\prec H_{j}$ if $|H_{i}|\leq|H_{j}|$. To apply Lemma 2.2, we need to associate each member $A\in\mathcal{A}$ with a set $X$ and at most $k-1$ many intersection conditions as follows.

• •

  For $H\in\mathcal{H}$, we can set $X:=H$ with intersection conditions $(\{h\},0)$ for each $h\in H$ and $([n],n-k-1)$.

• •

  For $F\in\mathcal{F}_{1}$, we can set $X:=F\setminus\{p\}$ with intersection conditions $(F\setminus\{p\},\beta)$ for $0\leq\beta\leq k-2$.

• •

  For $G\in\mathcal{G}$, we can set $X:=G$ with intersection conditions $(\{g\},0)$ for each $g\in G$.

• •

  For $S\in\mathcal{S}$, we can set $X:=S$ with intersection conditions $(\{s\},0)$ for each $s\in S$.

We claim that the system $(A_{i},X_{i},R_{i})$, in which $A_{i}\in\mathcal{A}$, $X_{i}$ and the intersections conditions $R_{i}$ are defined as above, satisfies the conditions in Lemma 2.2 with $s=k-1$.

Then by Lemma 2.2 and 3.1, we have $|\mathcal{H}|+|\mathcal{F}_{1}|+|\mathcal{G}|+|\mathcal{S}|\leq\sum\limits_{i=0}^{k-1}\binom{n}{i}$. Also note that $|\mathcal{G}|=|\mathcal{H}|=\sum\limits_{i=0}^{k-2}\binom{n-1}{i}$, which implies

$$d_{\textup{max}}(\mathcal{F})=|\mathcal{F}_{1}|\leq\sum\limits_{i=0}^{k-1}\binom{n}{i}-2\sum\limits_{i=0}^{k-2}\binom{n-1}{i}-|\mathcal{S}|=\binom{n-1}{k-1}-|\mathcal{S}|.$$

Moreover, we have

$$|\mathcal{F}|=|\mathcal{F}_{0}|+|\mathcal{F}_{1}|\leq\binom{n-1}{k-1}-(|\mathcal{S}|-|\mathcal{F}_{0}|).$$

We can immediately derive the following results when $|\mathcal{F}_{0}|$ is very small.

1. 1.

   When $|\mathcal{F}_{0}|=0$, suppose that there exists some $S\in\mathcal{S}$, then $S\cup\{p\}\notin\mathcal{F}$ and $(S\cup\{p\})\cap F\neq\emptyset$ for any $F\in\mathcal{F}$, which is a contradiction to that $\mathcal{F}$ is a maximal non-trivial intersecting family. Therefore, when $\mathcal{F}_{0}=\emptyset$, then $\mathcal{S}=\emptyset$, which gives $|\mathcal{F}|\leq\binom{n-1}{k-1}$.

2. 2.

   When $|\mathcal{F}_{0}|=1$, set $\mathcal{F}_{0}=\{F_{0}\}$. By definitions, for each $S\in\mathcal{S}$, we can see $S\cap F_{0}=\emptyset$ and $p\notin S$. Moreover, we claim that $\binom{[n]\setminus(\{p\}\cup F_{0})}{k-1}\subseteq\mathcal{S}$. To see this, since $\mathcal{F}$ is an intersecting family, then for any member $P\in\partial_{k-1}\mathcal{F}_{1}$, at least one of the events $p\in P$ and $P\cap F_{0}\neq\emptyset$ occurs, which yields that $\partial_{k-1}\mathcal{F}_{1}\cap\binom{[n]\setminus(F_{0}\cup\{p\})}{k-1}=\emptyset$. Therefore, when $|\mathcal{F}_{0}|=1$, then $|\mathcal{S}|\geq\binom{n-k-1}{k-1}$, which also yields that $|\mathcal{F}|\leq\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$.

In general, Let $x:=|\mathcal{F}_{0}|\geq 1$, and $\mathcal{F}_{0}:=\{F_{1},F_{2},\ldots,F_{x}\}$, we have the following crucial claim by extending the above argument in the case of $|\mathcal{F}_{0}|=1$.

For $\mathcal{F}_{0}=\{F_{1},F_{2},\ldots,F_{x}\}$, let $\mathcal{C}_{1}=\{C\subseteq\binom{[n]}{k-1}:p\notin C,C\cap F_{1}=\emptyset\}$. Moreover, for each $2\leq i\leq x$, we define

$$\mathcal{C}_{i}:=\bigg{\{}C\subseteq\binom{[n]}{k-1}:p\notin C,C\cap F_{i}=\emptyset,C\cap F_{j}\neq\emptyset\ \textup{for\ any\ }1\leq j\leq i-1\bigg{\}}.$$

By definitions one can see that $\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{x}$ are pairwise disjoint, moreover, it is not hard to see that $\bigcup\limits_{i=1}^{x}\mathcal{C}_{i}=\bigcup\limits_{i=1}^{x}\binom{[n]\setminus(F_{i}\cup\{p\})}{k-1}$. When $1\leq i\leq x\leq k$, by definition we have $|\mathcal{C}_{i}|\geq\binom{n-k-i}{k-i}$, therefore, we have the following lower bound on $|\mathcal{S}|$ by 3.2.

By 3.2 and 3.3, we can see $|\mathcal{S}|\geq\sum\limits_{j=1}^{k}\binom{n-k-j}{k-j}$ when $x>k$. Indeed, we can strengthen 3.3 in the following form, which plays a key role when we analyze the extremal structures. Recall that a family $\mathcal{W}$ is called a sunflower with $s$ common elements, if there is a set $S\subseteq[n]$ with $|S|=s$ such that for any distinct $W_{i},W_{j}\in\mathcal{W}$, we have $W_{i}\cap W_{j}=S$.