Erdős-Ko-Rado type theorems for simplicial complexes

It is shown that every shifted simplicial complex ∆ is EKR of type (r, s), provided that the size of every facet of ∆ is at least (2s + 1)r − s. It is moreover proven that every i-near-cone simplicial complex is EKR of type (r, i) if depthK∆ > (2i + 1)r − i − 1, for some field K. Furthermore, we prove that if G is a graph having at least (2i + 1)r − i connected components, including i isolated vertices, then its independence simplicial complex ∆G is EKR of type (r, i). The results of this paper, generalize the main result of Frankl (2013).


Introduction and preliminaries
Throughout this paper, the set of positive integers {1, 2, . . .} is denoted by N. For m, n ∈ N with m n, the set {i ∈ N : m i n} is denoted by [m, n]; for m = 1, we also write [n].
A family A of sets is intersecting if A ∩ B = ∅, for every pair of sets A, B ∈ A. A classical result in extremal set theory is the famous theorem of Erdős, Ko, and Rado [7]. It asserts that the maximum size of an intersecting family of r-subsets (i.e., subsets of size r) of [n] is n−1 r−1 , provided that n 2r. In other words, the largest possible intersecting families of r-subsets of [n] are the families of all r-subsets containing some fixed element of [n], whenever n 2r. For a nice survey on this topic we refer to [4].
Let A be a family of subsets of [n]. A subfamily M of A is called a matching if the elements of M are pairwise disjoint. The matching number of A, denoted by ν(A) is defined to be the size of the largest matching of A. Therefore, a nonempty family A is intersecting if and only if ν(A) = 1. In [6], Erdős proposed the following conjecture concerning the maximum size of a family of subsets of [n] which has a given matching number.
Conjecture 1.1. Assume that A is a family of r-subsets of [n] with ν(A) s. If n (s + 1)r − 1, then |A| max (s + 1)r − 1 r , n r − n − s r . (1) For s = 1, Conjecture 1.1 is the same as Erdős-Ko-Rado Theorem (note that Conjecture 1.1 is trivially true for s = 1 and n = r(s + 1) − 1). Also, for r = 1, the conjecture clearly holds. Erdős [6] proved that there exists an integer n 0 (r, s) such that the inequality (1) is valid for every family of r-subsets of [n] with ν(A) s, provided that n n 0 (r, s). Frankl [8] proved that n 0 (r, s) can be chosen to be (2s + 1)r − s. More explicit, he proved the following result. The aim of this paper is to extend Theorem 1.2 to some classes of simplicial complexes. Erdős-Ko-Rado type theorems for simplicial complexes have been studied by several authors; see e.g., [1], [2], [3], [12], [14]. Let us continue with some preliminaries from simplicial complexes.
A simplicial complex ∆ on the ground set V (∆) := [n] is a collection of subsets of [n] that is closed under taking subsets; that is, if F ∈ ∆ and F ⊆ F , then also F ∈ ∆. Every element F ∈ ∆ is called a face of ∆. The dimension of a face F is defined to be dim F := |F | − 1. The dimension of ∆, which is denoted by dim ∆, is defined to be d − 1, where d = max{|F | : F ∈ ∆}. A facet of ∆ is a maximal face of ∆ with respect to inclusion. Let F(∆) denote the set of facets of ∆. It is clear that F(∆) determines ∆. When F(∆) = {F 1 , . . . , F m }, we write ∆ = F 1 , . . . , F m and say that ∆ is generated by F 1 , . . . , F m . A simplicial complex ∆ is called pure if all facets of ∆ have the same size. We say that ∆ is a simplex if it consists of all subsets of [n]. Thus, a simplex has exactly one facet, namely [n]. The link of ∆ with respect to a subset F ⊆ [n], denoted by lk ∆ F , is the simplicial complex Let ∆ be a simplicial complex. The simplicial complex ∆ (i) := {F ∈ ∆ : dim F i} is the i-skeleton of ∆. Also, the simplicial complex ∆ [i] := F ∈ ∆ : dim F = i is the i-pure skeleton of ∆.
A face of ∆ of size r is an r-face of ∆. We denote the number of r-faces of ∆ by f r (∆).
the electronic journal of combinatorics 24(2) (2017), #P2.38 Note. Many authors define an r-face to be a face with dimension r. We follow Swartz [13] and Woodroofe [14] in considering an r-face to be a face with size r (rather than dimension r).
A simplicial complex ∆ is said to be Cohen-Macaulay over a field K if for every F ∈ ∆ and every i less than dim(lk ∆ F ), it holds that H i (lk ∆ F ; K) = 0, where H i (∆; K) denotes the simplicial homology of ∆ with coefficients in K (this definition of Cohen-Macaulayness coincides with the one which appears in the context of combinatorial commutative algebra via the Stanley-Reisner correspondence). It is well-known that every Cohen-Macaulay simplicial complex is pure (see for example [9,Lemma 8.1.5]). We say that a simplicial complex ∆ is sequentially Cohen-Macaulay over a field K if every pure skeleton of ∆ is Cohen-Macaulay over K.
We note that depth K ∆ is at most the minimum facet dimension of ∆, and equality holds if ∆ is sequentially Cohen-Macaulay over K. Let ∆ be a simplicial complex and W be a subset of V (∆). The anti-star of ∆ with respect to W , denoted by ast ∆ W , is the simplicial complex When W = {v} is a singleton, we sometimes write ∆ \ v instead of ast ∆ W .
where the minimum is taken over all subsets W of V (∆) with |W | = s.
We restate Theorem 1.2 using the language of simplicial complexes: Theorem 1.4. Let ∆ be a simplex on ground set [n]. If n (2s + 1)r − s, then ∆ is EKR of type (r, s).
In the subsequent sections, we extend Theorem 1.4 as follows. In Section 2, we focus on shifted simplicial complexes (see Definition 2.1). The main result of that section is Theorem 2.2, which asserts that every shifted simplicial complex ∆ is EKR of type (r, s), provided that the size of every facet of ∆ is at least (2s + 1)r − s. Our main tool in the proof of Theorem 2.2 is (exterior) algebraic shifting. This method was used in [12] and [14] to prove other Erdős-Ko-Rado type theorems for simplicial complexes. In Section 3, we consider i-near-cone simplicial complexes (see Definition 3.1). We show in Theorem 3.3 that every i-near-cone simplicial complex is EKR of type (r, i) if depth K ∆ (2i + 1)r − i − 1, for some field K. In Section 4, we concentrate on the independence simplicial complexes associated to graphs. In Proposition 4.2, we characterize the graphs for which the independence complex ∆ G is i-near-cone and conclude in Corollary 4.4 that if G is a graph having at least (2i + 1)r − i connected components, including i isolated vertices, then ∆ G is EKR of type (r, i).

Shifted simplicial complexes
In this section, using shifting theory, we study the property of being EKR of type (r, s) for shifted simplicial complexes. We first provide some definitions and basic facts from shifting theory.
Consider the set of all simplicial complexes on a given ordered ground set V . A shifting operation associates to each such simplicial complex ∆ a new simplicial complex Shift ∆ on the ground set V such that (S 1 ) For every simplicial complex ∆, Shift ∆ is a shifted complex.
If A is a family of r-subsets of [n], then Shift A is defined to be the set of r-faces of Shift(∆(A)), where ∆(A) is the simplicial complex generated by A. In our proofs we need a shifting operation that satisfies the following extra property: (S 5 ) If A has the property that among every s + 1 members of A, there are two with nonempty intersection, then the same property holds for Shift A. In other words, ν(Shift A) ν(A).
Kalai proved (see [10, Theorem 6.3 and subsequent Remarks]) that a specific shifting operation called exterior algebraic shifting (with respect to a field K) satisfies (S 5 ). We denote the exterior algebraic shift of ∆, with respect to a field K, by Shift K ∆. The precise definition of exterior algebraic shifting will not be important for us, but it can be found in [9] from a commutative algebraic perspective, or in [10] from a more elementary perspective.
We are now ready to prove that every shifted simplicial complex ∆ is EKR of type (r, s), provided that the size of facets of ∆ are large enough. In the proof we do not rely on a specific shifting operator, but only require (S 1 , S 2 , S 3 , S 4 , S 5 ) for the operator Shift.
Theorem 2.2. Let ∆ be a shifted complex having minimal facet size k. Then ∆ is EKR of type (r, s), for every natural numbers r and s with k (2s + 1)r − s.
Proof. Suppose that the ordered set {v 1 , . . . , v n } is the ground set of ∆. Let A be a family of r-faces of ∆ with ν(A) s. Note that by assumption n k > s.
the electronic journal of combinatorics 24(2) (2017), #P2.38 Let n be the size of the ground set of ∆. We proceed by induction on n. Note that if ∆ is a simplex, then Theorem 1.4 guarantees that the assertion is true. Also, there is nothing to prove if r = 1.
Thus assume that ∆ is not a simplex and r 2. It then follows from (S 1 ), (S 4 ) and (S 5 ) that Shift A is a shifted family of r-faces of Shift ∆ with ν(Shift A) s. On the other hand, ∆ is a shifted complex, and thus we conclude from (S 2 ) that Shift ∆ = ∆. This argument shows that Shift A is in fact a shifted family of r-faces of ∆, and (S 3 ) shows that its size is |A|. Let A 1 be the set of all faces σ ∈ Shift A with v n ∈ σ and set A 2 = Shift A \ A 1 . We study the size of A 1 and A 2 .
To study the size of A 1 , we set Hence |A 1 | = |A 1 |. We claim that ν(A 1 ) s.
Proof of the claim. Suppose by contradiction that ν(A 1 ) s + 1. Hence, there exist σ 1 , σ 2 , . . . , σ s+1 ∈ A 1 such that σ i ∩ σ j = {v n }, for every pair of integers i = j. It follows that for every j with 1 j s. Also, set σ s+1 = σ s+1 . By the definition of shiftedness, we know that σ j belongs to Shift A, for every j with 1 j s + 1. Further, σ i ∩ σ j = ∅, for every i = j. This shows that σ 1 , . . . , σ s+1 is a matching of Shift A. Hence, ν(Shift A) s + 1 which is a contradiction. This completes the proof of the claim.
It follows that A 1 is a family of (r − 1)-faces of lk ∆ v n with ν(A 1 ) s. Notice lk ∆ v n is a shifted complex on ground set {v 1 , . . . , v n−1 }. It is clear that the minimum facet size of lk ∆ v n is at least k − 1 and k − 1 (2s + 1)(r − 1) − s. On the other hand, r 2 and this shows that the size of the ground set of lk ∆ v n is at least k − 1 s + 1. Thus, W is a subset of the ground set of lk ∆ v n . The induction hypothesis implies that We now consider A 2 . It is clear that A 2 is a a family of r-faces of ∆\v n with ν(A 2 ) s. Since ∆ is a shifted complex which is not a simplex, we conclude that the minimum facet size of ∆ \ v n is at least k. Hence the induction hypothesis implies that Finally we have the electronic journal of combinatorics 24(2) (2017), #P2.38 The desired inequality now follows by observing that

Intersecting faces of i-near-cones
In this section we study the property of being EKR of type (r, s) for i-near-cone simplicial complexes. The main result of this section is Theorem 3.3 which states that an i-nearcone is EKR of type (r, i), provided that depth K ∆ is large enough. In the proof, we use exterior algebraic shifting and Theorem 2.2. Therefore in this section we fix a field K and by Shift ∆ we always mean the exterior algebraic shifting with respect to K. Let ∆ be a simplicial complex and v be a member of the ground set of ∆. The complex ∆ is a near-cone with respect to the apex vertex v if (σ \{w})∪{v} is a face of ∆ whenever σ is a face of ∆ and w is an element of σ.
We next define the notion of i-near cone simplicial complexes. It was first introduced by Nevo [
Let ∆ be a simplicial complex with ground set V (∆). For every subset G ⊆ V (∆), the restriction of ∆ to G is defined to be the simplicial complex The following proposition has a crucial role in the proof of Theorem 3.3.
the electronic journal of combinatorics 24(2) (2017), #P2.38 Proposition 3.2. Let ∆ be an i-near cone. Assume that v 1 , . . . , v i is the apex of ∆ and set F = {v 1 , . . . , v i }. Suppose that the minimum size of facets of ∆ and Shift ∆ is at least k with k i. View Shift ∆ as having ordered ground set {u 1 , . . . , u n }, and view Shift(ast ∆ F ) as having ordered ground set G = {u i+1 , . . . , u n }. For r k − 2i, Proof. Consider a face σ ∈ (Shift ∆) G , with |σ| = r. Since the minimum facet size of Shift ∆ is at least k and r k − 2i < k − i, we conclude that Shift ∆ has a face of size r + i containing σ. It then follows from the definition of shiftedness that Thus σ ∈ lk Shift ∆ {u 1 , . . . , u i }. This shows that The converse inequality is trivial, because Hence we conclude that Now consider a face τ ∈ ast ∆ F with |τ | = r. We use the notation from Definition 3.1 to prove that τ ∪ F ∈ ∆. Since the minimal facet size of ∆ is at least k r + 2i > r + i, there exists w ∈ V (∆) \ (τ ∪ F ) such that τ ∪ {w} ∈ ∆. It is clear from Definition 3.1 that τ ∪ {w} ∈ ∆(i) ⊂ ∆(i − 1). Since ∆(i − 1) is a near-cone with respect to v i , we conclude that Let j be the least integer such that τ ∪ {v j , . . . , v i } ∈ ∆. We should prove that j = 1. Assume by contradiction that j > 1. Again, since the minimal facet size of ∆ is at least It is clear that Since ∆(j − 2) is a near-cone with respect to v j−1 , we conclude that which contradicts the choice of j. Therefore τ ∪ F ∈ ∆, which yields that τ ∈ lk ∆ F . This shows that f r (ast ∆ F ) f r (lk ∆ F ).
the electronic journal of combinatorics 24(2) (2017), #P2.38 The converse inequality is trivial, because lk ∆ F ⊆ ast ∆ F . Hence, using S 3 , we conclude that On the other hand, we know from [12, Proposition 3.6] that The assertion now follows from the above equality together with equalities † and ‡.
We are now ready to prove the main result of this section.
Theorem 3.3. Let r and i be positive integers and ∆ be an i-near-cone with Then ∆ is EKR of type (r, i).
Proof. The case r = 1 is trivial. Hence, suppose that r 2. Assume that v 1 , . . . , v i is the apex of ∆ and set F = {v 1 , . . . , v i }. Consider a family A of r-faces of ∆ with ν(A) s. It suffices to prove that |A| f r (∆) − f r (ast ∆ F ). In order to do this, we use algebraic shifting. Consider the simplicial complex Shift ∆ and assume that the ordered set {u 1 , . . . , u n } is its ground set. It follows from (S 5 ) that Shift A a family of r-faces of Shift ∆ with ν(Shift A) s. It also follows from (S 3 ) that its size is | Shift A| = |A|. Let k be the minimum size of facets of ∆ and Shift ∆. By [5,Corollary 4.5] and the definition of depth, we conclude that k depth K ∆ + 1 (2i + 1)r − i > r + 2i.
Set W = {u 1 , . . . , u i } and G = {u i+1 , . . . , u n }. It follows from (S 3 ), Remark 2.3 and Proposition 3.2 that The following corollary is an immediate consequence of Theorem 3.3 and proves that every sequentially Cohen-Macaulay i-near-cone ∆ is EKR of type (r, i), provided that the size of every facet of ∆ is large enough. Note that if ∆ is sequentially Cohen-Macaulay over K then depth K ∆ is the minimum facet dimension of ∆.
We recall that A ⊆ V (G) is an independent set in G if none of its elements are adjacent. We say that a graph G is sequentially Cohen-Macaulay (resp. near-cone, i-near-cone) if its independence simplicial complex ∆ G is sequentially Cohen-Macaulay (resp. near-cone, i-near-cone). In this section, we characterize the i-near-cone graphs. We first consider the near-cone case. The following lemma says that a graph G with n vertices is near cone if and only if it has a vertex v such that every neighborhood of v has degree n − 1 Proof. Notice that ∆ G is near-cone with apex v if and only if (A \ {w}) ∪ {v} is an independent set of G, for every independent set A ∈ ∆ G and every vertex w ∈ A. This equivalent to say that v is not adjacent to any vertex of A \ {w}, for every independent set A ∈ ∆ G and every vertex w ∈ A. This means that v is not adjacent to any vertex of A, for every independent set A ∈ ∆ G with |A| 2. Therefore, ∆ G is near-cone with apex vertex v if and only if for every vertex u ∈ N G (v), the largest independent set of G, containing u is the singleton {u}, i.e., N G [u] = V (G).
The following proposition is an immediate consequence of Lemma 4.1 and characterizes i-near-cone graphs. Recall that for every graph G and every vertex v ∈ V (G), the graph G \ v is obtained from G by deleting the vertex v and every edge adjacent to v. Proposition 4.2. A graph G is an i-near-cone if and only if there are distinct vertices v 1 , . . . , v i ∈ V (G) and a chain of subgraphs G(0) ⊃ G(1) ⊃ · · · ⊃ G(i) with G(0) = G such that whenever 1 j i the following conditions hold.
In particular, G is an i-near-cone if it has i isolated vertices.
For a graph G, we denote the minimal facet size of ∆ G by minind(G). It is clear that minind(G) is equal to the minimum size of maximal independent sets of G. As an immediate consequence of Corollary 3.4 and Proposition 4.2, we conclude the following result.
A list of sequentially Cohen-Macaulay graphs can be found in [14, Page 1224[14, Page -1225. In particular every chordal graph (i.e., the graph which has no induced cycle of length at least 4) is sequentially Cohen-Macaulay. Also, if every connected component of G is sequentially Cohen-Macaulay, then G is sequentially Cohen-Macaulay too.
It is know by [14,Lemma 2.12] that for every field K, we have depth K ∆ 1 * ∆ 2 = depth K ∆ 1 + depth K ∆ 2 + 1. On the other hand one can easily see that if G is a graph with connected components G 1 , . . . , G t , then ∆ G = ∆ G 1 * · · · * ∆ Gt . This yields that for every field K and every graph G with t connected components, the quantity depth K ∆ G is at least t − 1.
Corollary 4.4. Let r and i be positive integers and G be a graph having i isolated vertices. If the number of connected components of G is at least (2i + 1)r − i, then ∆ G is EKR of type (r, i).
Proof. The assertion follows immediately from Theorem 3.3, Proposition 4.2 and the above argument.
Let G be a graph. The complementary graph G is the graph with V (G) = V (G) and E(G) consists of those 2-element subsets {u, v} of V (G) for which {u, v} / ∈ E(G). Woodroofe [14,Lemma 4.4] proves that if G is graph with at least two vertices which has connected complement, then for every field K, we have depth K ∆ G 1. As a consequence, we conclude the following result.
Corollary 4.5. Let r, i, m and t be positive integers and G be a graph with connected components G 1 , . . . , G t . Assume further that (i) t (2i + 1)r − i − m, (ii) G 1 , . . . , G i are isolated vertices and (iii) For every integer j with i + 1 j i + m, the graph G j has at least two vertices and the complementary graph G j is connected.
Then ∆ G is EKR of type (r, i).