The flag upper bound theorem for 3- and 5-manifolds

We prove that among all flag 3-manifolds on $n$ vertices, the join of two circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and $\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and find all maximizers of the face numbers in this class.


Introduction
One of the classical problems in geometric combinatorics deals with the following question: for a given class of simplicial complexes, find tight upper bounds on the number of i-dimensional faces as a function of the number of vertices and the dimension. Since Motzkin (1957) proposed the upper bound conjecture (UBC, for short) for polytopes in 1957, this problem has been solved for various families of complexes. In particular, McMullen (1970) and Stanley (1975) proved that neighborly polytopes simultaneously maximize all the face numbers in the class of polytopes and simplicial spheres. However, it turns out that, apart from cyclic polytopes, many other classes of neighborly spheres or even neighborly polytopes exist, see Shermer (1982) and Padrol (2013) for examples and constructions of neighborly polytopes.
A simplicial complex ∆ is flag if all of its minimal non-faces have cardinality two, or equivalently, ∆ is the clique complex of its graph. Flag complexes form a beautiful and important class of simplicial complexes. For example, barycentric subdivisions of simplicial complexes, order complexes of posets, and Coxeter complexes are flag complexes. Despite a lot of effort that went into studying the face numbers of flag spheres, in particular in relation with the Charney-Davis conjecture (Charney and Davis, 1995), and its generalization given by Gal's conjecture (Gal, 2005), a flag upper bound theorem for spheres is still unknown. The upper bounds of face numbers for general simplicial (d−1)-spheres are far from sharp for those of flag (d − 1)-spheres, since the graph of any flag (d − 1)-dimensional complex is K d+1 -free. Gal (2005) established the upper bound theorem for flag spheres of dimension less than five. However, starting from dimension five, there are only conjectural upper bounds. For m ≥ 1, we let J m (n) be the (2m − 1)-sphere on n vertices obtained as the join of m copies of the circle, each one a cycle with either n m or n m vertices. We also let J * m (n) be the 2m-sphere on n vertices defined as the suspension of J m (n − 2). Conjecture 1.1 (Nevo and Petersen, 2010, Conjecture 6.3) If ∆ is a flag homology sphere, then γ(∆) satisfies the Frankl-Füredi-Kalai inequalities on dim ∆+1 2 -colored complexes. In particular, if ∆ is of As for the case of equality, Lutz and Nevo posited that as opposed to the case of all simplicial spheres, for a fixed dimension 2m − 1 and the number of vertices n, there is only one maximizer of the face numbers.
Recently, Adamaszek and Hladký (2015) proved that this conjecture holds asymptotically for flag homology manifolds. Several celebrated theorems from extremal graph theory served as tools for their work. As a result, the proof simultaneously gives upper bounds on f -numbers, h-numbers, g-numbers and γnumbers, but it only applies to flag homology manifolds with an extremely large number of vertices.
Our first main result is that both Conjecture 1.1 and Conjecture 1.2 hold for all flag 3-manifolds.
The proof of Theorem 1.3 only relies on simple properties of flag complexes and Eulerian complexes. We also establish an analogous result on the number of edges of flag 5-manifolds.
In 1964, Klee (1964) proved that Motzkin's UBC for polytopes holds for a much larger class of Eulerian complexes as long as they have sufficiently many vertices, and conjectured that the UBC holds for all Eulerian complexes. Our second main result deals with flag Eulerian complexes, and asserts that Conjecture 1.1 continues to hold for all 3-dimensional flag Eulerian complexes.
Theorem 1.5 Let ∆ be a 3-dimensional flag Eulerian complex on n vertices.
This provides supporting evidence to a question of Adamaszek and Hladký (2015, Problem 17(i)) in the case of dimension 3, where they proposed that Conjecture 1.1 holds for all odd-dimensional flag weak pseudomanifolds with sufficiently many vertices. We also give constructions of the maximizers of face numbers in this class and show that they are the only maximizers. Our proof is based on an application of the inclusion-exclusion principle and double counting.
The Extended Abstract is organized as follows. In Section 2, we discuss basic facts on simplicial complexes and flag complexes. In Section 3, we provide the proof of our first main result asserting that given a number of vertices n, the maximum face numbers of a flag 3-manifold are achieved only when this manifold is the join of two circles of length as close as possible to n 2 . In Section 4, we apply an analogous argument to the class of flag 5-manifolds. In Section 5, we show that the same upper bounds continue to hold for the class of 3-dimensional flag Eulerian complexes, and discuss the maximizers of the face numbers in this class. Finally, we close in Section 6 with some concluding remarks.

Preliminaries
A simplicial complex ∆ on a vertex set V = V (∆) is a collection of subsets σ ⊆ V , called faces, that is closed under inclusion. For σ ∈ ∆, let dim σ := |σ| − 1 and define the dimension of ∆, dim ∆, as the maximal dimension of its faces. A facet in ∆ is a maximal face under inclusion, and we say that ∆ is pure if all of its facets have the same dimension.
A simplicial complex ∆ is a simplicial manifold (resp. simplicial sphere) if the geometric realization of ∆ is homeomorphic to a manifold (resp. sphere). We denote byH * (∆; k) the reduced homology of ∆ computed with coefficients in a field k, and by β i (∆; k) := dim kHi (∆; k) the reduced Betti numbers of ∆ with coefficients in k. We say that ∆ is a (d − 1)-dimensional k-homology manifold if H * (lk ∆ σ; k) ∼ =H * (S d−1−|σ| ; k) for every nonempty face σ ∈ ∆. A k-homology sphere is a k-homology manifold that has the k-homology of a sphere. Every simplicial manifold (resp. simplicial sphere) is a homology manifold (resp. homology sphere). Moreover, in dimension two, the class of homology 2spheres coincides with that of simplicial 2-spheres, and hence in dimension three, the class of homology 3-manifolds coincides with that of simplicial 3-manifolds. For be the reduced Euler characteristic of ∆. A simplicial complex ∆ is called an Eulerian complex if ∆ is pure and χ(lk ∆ σ) = (−1) dim lk∆ σ for every σ ∈ ∆, including σ = ∅. In particular, it follows from the Poincaré duality theorem that all odd-dimensional simplicial manifolds are Eulerian. A it is connected, and the link of each face of dimension ≤ d−3 is also connected. Every Eulerian complex is a weak pseudomanifold, and every connected homology manifold is a normal pseudomanifold. In fact, every normal 2-pseudomanifold is also a homology 2manifold. However, for d > 3, the class of normal (d − 1)-pseudomanifolds is much larger than the class of homology (d − 1)-manifolds. It is well-known that if ∆ is a weak (resp. normal) (d − 1)pseudomanifold and σ is a face of ∆ of dimension at most d − 2, then the link of σ is also a weak (resp. normal) pseudomanifold. The following lemma gives another property of normal pseudomanifolds, see Bagchi and Datta (2008, Lemma 1.1).
Then the induced subcomplex of ∆ on vertex set V (∆)\W has at most two connected components.
Since the graph of any simplicial 2-sphere is a maximal planar graph, it follows that the f -vector of a simplicial 2-sphere is uniquely determined by f 0 . For a 3-dimensional Eulerian complex, the following lemma indicates that its f -vector is uniquely determined by f 0 and f 1 . (We omit the proof.) A simplicial complex ∆ is flag if all minimal non-faces of ∆, also called missing faces, have cardinality two; equivalently, ∆ is the clique complex of its graph. The following lemma (Nevo and Petersen, 2010, Lemma 5.2) gives a basic property of flag complexes: In particular, all links in a flag complex are also flag.
Finally, we recall some terminology from graph theory. A graph G is a path graph if the set of its vertices can be ordered as x 1 , x 2 , · · · , x n in such a way that {x i , x i+1 } is an edge for all 1 ≤ i ≤ n − 1 and there are no other edges. Similarly, a cycle graph is a graph obtained from a path graph by adding an edge between the end points of the path.

The Proof of flag UBC for flag 3-manifolds
Recall that in the introduction, we defined J m (n) to be the (2m − 1)-sphere on n vertices obtained as the join of m circles, each one of length either n m or n m . The goal of this section is to prove the flag UBC for flag 3-manifolds (see Conjectures 1.1 and 1.2). We start with the following lemma.
By applying the above argument inductively, we obtain that ∆[W 2 \{v}] is a path graph u 1 , u 2 , · · · , u n−a−1 , and there is a vertex z 2 in W 1 such that lk ∆ u 1 = {z 2 , u 2 } * C 1 and lk ∆ v = C 1 * {z 1 , z 2 }. Furthermore, C 1 ⊆ lk ∆ u i for all u i ∈ W 2 . Then we let C 2 be the cycle graph (v, z 2 , u 1 , u 2 , · · · , u n−a−1 , z 1 ). It follows that ∆ = C 1 * C 2 . Since a = |C 1 | + 2 = n+4 2 or n+4 4 , C 1 and C 2 must be cycles of length n 2 or n 2 . This implies ∆ = J 2 (n). By Lemma 2.2, the value of f 2 or f 3 determines f 1 , and if either of them is maximal, then also f 1 is maximal. This yields the result. 2

Counting edges of flag homology 5-manifolds
Recall that we use J * m (n) to denote the suspension of J m (n − 2). For even-dimensional flag homology spheres, the following is a special case of the last part of Conjecture 1.1: Conjecture 4.1 Fix m ≥ 1. For every flag homology 2m-sphere ∆ on n vertices, we have f 1 (M ) ≤ f 1 (J * m (n)). Using the techniques similar to those in Section 3, we establish the following proposition. In this section, we find tight upper bounds on the face numbers for all 3-dimensional flag Eulerian complexes. The proof relies on the following three lemmas.
(a) If σ 1 and σ 2 are two ridges that lie in the same facet σ in ∆, then the links of σ 1 and σ 2 are disjoint.
The proof follows from the definition of flag complexes. We omit it for the sake of brevity. Lemma 5.1 part (b) implies that if ∆ is a flag 3-dimensional simplicial complex and σ ∈ ∆ is a facet, then e⊆σ f 0 (lk ∆ e) ≤ 3f 0 (∆), where the sum is over the edges of σ. The following lemma suggests a better estimate on e⊆σ f 0 (lk ∆ e) for an arbitrary flag weak 3-pseudomanifold ∆.
Also since ∆ is a weak 3-pseudomanifold, any ridge of ∆ is contained in exactly two facets.
is a set of cardinality two. By the inclusiong-exclusion principle, we obtain that For simplicity, we denote the set (V 1 ∪ V 2 ) ∩ (V 3 ∪ V 4 ) asV . Notice that by Lemma 5.1 part (b), any vertex v ∈ ∆ belongs to at most one of the sets V 1 ∩ V 2 and V 3 ∩ V 4 . We split the vertices of ∆ into the following three types.
∈V . Each of these vertices contributes 1 to the right-hand side of (2).

If
v ∈V . By Lemma 5.1 part (a), every pair of ridges in σ has disjoint links. Since |V i ∩ V j ∩ V k | = 2, the number of such vertices is exactly 8, and each of them contributes 2 to the right-hand side of (2).
3. If v / ∈ V 1 ∩ V 2 and v / ∈ V 3 ∩ V 4 , then v contributes to the right-hand side of (2) at most 1. This case occurs only when v ∈V , that is, when v belongs to one of V 1 and V 2 , and one of V 3 and V 4 .
Proof: The proof uses the inclusion-exclusion principle and Lemma 5.2. 2 Now we are ready to prove the main result of this section.
Proof of Theorem 1.5: We denote the vertices of ∆ by v 1 , v 2 , · · · , v n and we let a i = f 0 (lk ∆ v i ). Since lk ∆ v i is an Eulerian complex of dimension 2, the f -numbers of lk ∆ v i satisfy the relations Hence f 2 (lk ∆ v i ) = 2a i − 4. By double counting, we obtain that σ∈∆,|σ|=4 v∈σ By Lemma 5.3, the left-hand side of (3) is bounded above by f 3 (∆)(2n + 8), which also equals (f 1 (∆) − n)(2n + 8) by Lemma 2.2. However, since 2f 1 (∆) = n i=1 f 0 (lk ∆ v 1 ), the right-hand side of (3) is bounded below by n · 2f1(∆) n · 4f1(∆) n − 4 , and equality holds only if a i = 2f1(∆) n for all 1 ≤ i ≤ n. Hence, We simplify this inequality to get + n, then there must be n 2 vertices such that f 0 (lk ∆ v) = n 2 + 2, while the rest of vertices have f 0 (lk ∆ v) = n 2 + 2. This proves our claim. 2 The following corollary provides some further properties of the maximizers of the face numbers in the class of 3-dimensional flag Eulerian complexes. (We omit the proof.) Corollary 5.4 Let ∆ be a 3-dimensional flag Eulerian complex on n vertices. If f 1 (∆) = f 1 (J 2 (n)), then ∆ and all of its vertex links are connected.
The next lemma provides a sufficient condition for a flag complex to be the join of two of its links. The proof simply relies on Lemma 2.3 and properties of normal pseudomanifolds; we omit it here.
Remark 5.6 The second result in Lemma 5.5 does not hold for flag weak pseudomanifolds, even assuming connectedness. Indeed, let L 1 , · · · , L 4 be four distinct circles of length ≥ 4. Then ∆ = (L 1 * L 3 ) ∪ (L 2 * L 3 ) ∪ (L 1 * L 4 ) is a flag weak 3-pseudomanifold. If τ 1 and τ 2 are edges in L 1 and L 3 respectively, then In Theorem 1.3, we proved that the maximizer of the face numbers is unique in the class of flag 3manifolds on n vertices. Is this also true for 3-dimensional flag Eulerian complexes? Corollary 5.4 implies that if the case of equality is not a join of two circles, then some of its edge links are not connected. Motivated by the example in Remark 5.6, we construct a family of 3-dimensional flag Eulerian complexes on n vertices that have the same f -numbers as those of J 2 (n).
Example 5.7 We write C n to denote a circle of length n. Let a 1 , a 2 , · · · , a s , b 1 , b 2 , · · · , b t ≥ 4 be integers such that 1≤i≤s a i = n 2 , and We claim that ∆ = ∪ 1≤i≤s,1≤j≤t (C ai * C bj ) is flag and Eulerian, where all C · are defined on disjoint vertex sets. Since the circles C ai and C bj are of length ≥ 4, it follows that ∆ is flag. Also any ridge τ in ∆ can be expressed as τ = {v} ∪ e, where v ∈ C ai and the edge e ∈ C bj (or v ∈ C bj and e ∈ C ai ) for some i, j. By the construction of ∆, the ridge τ is contained in exactly two facets {v, v } ∪ e and {v, v } ∪ e of ∆, where v and v are neighbors of v in the circle C ai (or C bj ). Hence the links of ridges in ∆ are Eulerian. Since the edge links in ∆ are either a circle or disjoint union of circles, and the vertex links in ∆ are suspensions of disjoint union of circles, these links are also Eulerian. Finally, the vertices in C ai have degree n 2 + 2 and the vertices in C bj have degree n 2 , and thus f 1 (∆) = f 1 (J 2 (n)). A simple computation also shows that f 2 (∆) = f 2 (J 2 (n)) and f 3 (∆) = f 3 (J 2 (n)). Hence χ(∆) = χ(J 2 (n)) = 0 and ∆ is Eulerian.
We denote the set of all complexes on n vertices constructed in Example 5.8 as GJ(n). It turns out that GJ(n) is exactly the set of maximizers of the face numbers in the class of flag 3-dimensional Eulerian complex on n vertices. To prove this, we begin with the following lemma.
Lemma 5.8 Let ∆ be a flag 3-dimensional Eulerian complex on n vertices. If f 1 (∆) = f 1 (J 2 (n)), then every vertex link is the suspension of disjoint union of circles.
Proof: By Lemma 5.8, we may assume that the link of vertex v 1 ∈ ∆ is the join of C and two other vertices v 2 , v 3 , where C is the disjoint union of circles. Then again by Lemma 5.8, the link of vertex v 2 is also the suspension of C. If v 1 is any vertex of C and its adjacent vertices in C are v 2 , v 3 , then by Lemma 2.3, ∆[V (C)] = C, and it follows that f 0 (lk ∆ v i ∩ C) = 2 for i = 1, 2. Hence for 1 ≤ i, j ≤ 2, f 0 (lk ∆ e) ≤ n + 4 · 4 = n + 16, where the sum is over the edges of {v 1 , v 2 , v 1 , v 2 }. Since f 1 (∆) = f 1 (J 2 (n)), by the proof of Theorem 1.5 and Lemma 5.2, it follows that this sum is exactly n + 16.
Remark 5.11 The complexes from Example 5.8 form asymptotically the complete list of maximizers of the number of edges in the class of K 1,3,3 -free graphs, see (Simonovits, 1966, Theorem 5). (Here K r1,r2,··· ,rm denotes the complete m-partite graph with r i vertices of color i.) A more general result on extremal graphs not containing K r1,r2,··· ,rm can be found in Erdös and Simonovits (1972). Studying these extremal graphs is the main tool of Adamaszek and Hladký's work Adamaszek and Hladký (2015) on asymptotic upper bounds.

Concluding Remarks
We close this paper with a few remarks and open problems. As mentioned in the introduction, Klee (1964) verified that the Motzkin's UBC for polytopes holds for Eulerian complexes with sufficiently many vertices, and conjectured it holds for all Eulerian complexes. Can the flag upper bounds for spheres also be extended to Eulerian complexes? Motivated by Theorem 1.3 and Theorem 1.5, we posit the following conjecture in the same spirit as Problem 17(i) from Adamaszek and Hladký (2015): Conjecture 6.1 Let ∆ be a flag (2m − 1)-dimensional complex, where m ≥ 2. Assume further that ∆ is an Eulerian complex on n vertices. Then f i (∆) ≤ f i (J m (n)) for all i = 1, · · · , 2m − 1.
Theorem 1.5 gives an affirmative answer in the case of m = 2 and 1 ≤ i ≤ 3. The next case is i = 1 and m = 3. In this case, Theorem 1.4 verifies Conjecture 6.1 for flag 5-manifolds. At present other cases are completely open.
The above results and conjectures discuss odd-dimensional flag complexes. What happens in the evendimensional cases? To this end, we pose the following strengthening of Conjecture 18 from Adamaszek and Hladký (2015).
Recall that J * m (n) = S 0 * C 1 * · · · * C m , where n ≥ 4m+2, each C i is a circle of length either n−2 m or n−2 m so that the total number of vertices of J * m (n) is n. Now we let S n denote the set of flag 2-spheres on n vertices, and define J * m (n) := {S * C 2 * · · · * C m | S ∈ S V (C1)+2 }.
It is not hard to see that every element in J * m (n) is a flag 2m-sphere. Conjecture 6.2 Let ∆ be a flag homology 2m-sphere on n vertices. Then f i (∆) ≤ f i (J * m (n)) for all i = 1, · · · , 2m. If equality holds for some 1 ≤ i ≤ 2m, then ∆ ∈ J * m (n).