Nowhere Dense Graph Classes and Dimension

Nowhere Dense Graph Classes and Dimension

Gwenaël Joret Computer Science Department
Université Libre de Bruxelles, Brussels, Belgium
Piotr Micek Theoretical Computer Science Department
Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland — and — Institute of Mathematics, Combinatorics and Graph Theory Group
Freie Universität Berlin, Berlin, Germany
Patrice Ossona de Mendez Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)
190-198 avenue de France, 75013 Paris, France — and — Computer Science Institute of Charles University (IUUK)
Malostranské nám. 25, 11800 Praha 1, Czech Republic
and  Veit Wiechert Institut für Mathematik
Technische Universität Berlin, Berlin, Germany
Abstract.

Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every and every , posets of height at most with elements and whose cover graphs are in the class have dimension .

G. Joret is supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. P. Micek is partially supported by a Polish National Science Center grant (SONATA BIS 5; UMO-2015/18/E/ST6/00299). G. Joret and P. Micek also acknowledge support from a joint grant funded by the Belgian National Fund for Scientific Research (F.R.S.–FNRS) and the Polish Academy of Sciences (PAN). V. Wiechert is supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408). P. Ossona de Mendez is supported by grant ERCCZ LL-1201 and by the European Associated Laboratory “Structures in Combinatorics” (LEA STRUCO)

1. Introduction

A class of graphs is nowhere dense if for every , there exists such that no graph in the class contains a subdivision of the complete graph where each edge is subdivided at most times as a subgraph. Examples of nowhere dense classes include most sparse graph classes studied in the literature, such as planar graphs, graphs with bounded treewidth, graphs excluding a fixed (topological) minor, graphs with bounded maximum degree, graphs that can be drawn in the plane with a bounded number of crossings per edges, and more generally graph classes with bounded expansion.

At first sight, being nowhere dense might seem a weak requirement for a graph class to satisfy. Yet, this notion captures just enough structure to allow solving a wide range of algorithmic problems efficiently: In their landmark paper, Grohe, Kreutzer, and Siebertz [7] proved for instance that every first-order property can be decided in almost linear time on graphs belonging to a fixed nowhere dense class.

One reason nowhere dense classes attracted much attention in recent years is the realization that they can be characterized in several, seemingly different ways. Algorithmic applications in turn typically build on the ‘right’ characterization for the problem at hand and sometimes rely on multiple ones, such as in the proof of Grohe et al. [7]. Nowhere dense classes were characterized in terms of shallow minor densities [22] and consequently in terms of generalized coloring numbers (by results from [30]), low tree-depth colorings [22] (by results from [19]), and subgraph densities in shallow minors [21]; they were also characterized in terms of quasi-uniform wideness [20, 16, 25], the so-called splitter game [7], sparse neighborhood covers [7], neighborhood complexity [3], the model theoretical notion of stability [1], as well as existence of particular analytic limit objects [18]. The reader is referred to the survey on nowhere dense classes by Grohe, Kreutzer, and Siebertz [6] for an overview of the different characterizations, and to the textbook by Nešetřil and Ossona de Mendez [23] for a more general overview of the various notions of sparsity for graphs (see also [24]).

The main contribution of this paper is a new characterization of nowhere dense classes that brings together graph structure theory and the combinatorics of partially ordered sets (posets). Informally, we show that the property of being nowhere dense can be captured by looking at the dimension of posets whose order diagrams are in the class when seen as graphs.

Recall that the dimension of a poset is the least integer such that the elements of can be embedded into in such a way that in if and only if the point of is below the point of with respect to the product order of . Dimension is a key measure of a poset’s complexity.

The standard way of representing a poset is to draw its diagram: First, we draw each element as a point in the plane, in such a way that if in the poset then is drawn below . Then, for each relation in the poset not implied by transitivity (these are called cover relations), we draw a -monotone curve going from up to . The diagram implicitly defines a corresponding undirected graph, where edges correspond to pairs of elements in a cover relation. This is the cover graph of the poset. Let us also recall that the height of a poset is the maximum size of a chain in the poset (a set of pairwise comparable elements).

Recall that a monotone class means a class closed under taking subgraphs. Our main result is the following theorem.

Theorem 1.

Let be a monotone class of graphs. Then is nowhere dense if and only if for every integer and real number , -element posets of height at most whose cover graphs are in have dimension .

This result is the latest step in a series of recent works connecting poset dimension with graph structure theory. This line of research began with the following result of Streib and Trotter [26]: For every fixed , posets of height with a planar cover graph have bounded dimension. That is, the dimension of posets with planar cover graphs is bounded from above by a function of their height. This is a remarkable theorem, because in general bounding the height of a poset does not bound its dimension, as shown for instance by the height- posets called standard examples, depicted in Figure 1 (left). Requiring the cover graph to be planar does not guarantee any bound on the dimension either, as shown by Kelly’s construction [13] of posets with planar cover graphs containing large standard examples as induced subposets (Figure 1, right). Thus, it is the combination of the two ingredients, bounded height and planarity, that implies that the dimension is bounded.

Soon afterwards, it was shown in a sequence of papers that requiring the cover graph to be planar in the Streib-Trotter result could be relaxed: Posets have dimension upper bounded by a function of their height if their cover graphs

• have bounded treewidth, bounded genus, or more generally exclude an apex-graph as minor [9];

• exclude a fixed graph as a (topological) minor [28, 17];

• belong to a fixed class with bounded expansion [12].

A class of graphs has bounded expansion if for every , there exists such that no graph in the class contains a subdivision of a graph with average degree at least where each edge is subdivided at most times as a subgraph. This is a particular case of nowhere dense classes.

Zhu [30] characterized bounded expansion classes as follows: A class has bounded expansion if and only if for every , there exists such that every graph in the class has weak -coloring number at most . Weak coloring numbers were originally introduced by Kierstead and Yang [14] as a generalization of the degeneracy of a graph (also known as the coloring number). They are defined as follows. Let be a graph and consider some linear order on its vertices (it will be convenient to see as ordering the vertices of from left to right). Given a path in , we denote by the leftmost vertex of w.r.t. . Given a vertex in and an integer , we say that is weakly -reachable from w.r.t.  if there exists a path of length at most from to in such that . We let denote the set of weakly -reachable vertices from w.r.t.  (note that this set contains for all ). The weak -coloring number of is defined as

 wcolr(G):=minπmaxv∈V(G)|WReachπr[v]|.

The novelty of our approach in this paper is that we bound the dimension of a poset using weak coloring numbers of its cover graph. Indeed, the general message of the paper is that dimension works surprisingly well with weak coloring numbers. We give a first illustration of this principle with the following theorem:

Theorem 2.

Let be a poset of height at most , let denote its cover graph, and let . Then

 dim(P)⩽4c.

To prove this, we first make the following observation about weak reachability.

Observation 3.

Let be a graph and let be a linear order on its vertices. If are vertices of such that is weakly -reachable from (w.r.t. ), is weakly -reachable from , and is a path from to in of length at most such that , then

one of is weakly -reachable from the other.

In particular, this holds if , i.e. .

Proof.

Consider a path from to witnessing that is weakly -reachable from , and a path from to witnessing that is weakly -reachable from . The union of , and contains a path connecting to of length at most . Since is the leftmost vertex of in and is the leftmost vertex of in , and we have that one of and is the leftmost vertex of in . This proves that one of is weakly -reachable from the other. ∎

Before continuing with the proof, let us introduce some necessary definitions regarding posets. Let be a poset. An element covers an element if in and there is no element such that in . A chain of is said to be a covering chain if the elements of can be enumerated as in such a way that covers in for each . (We use the notation .) The upset of an element is the set of all elements such that in . Similarly, the downset of an element is the set of all elements such that in . Note that and .

An incomparable pair of is an ordered pair of elements of that are incomparable in . We denote by the set of incomparable pairs of . Let be a non-empty set of incomparable pairs of . We say that is reversible if there is a linear extension of reversing each pair of , that is, we have in for every . We denote by the least integer such that can be partitioned into reversible sets. We will use the convention that when is an empty set. As is well known, the dimension of can equivalently be defined as , that is, the least integer such that the set of all incomparable pairs of can be partitioned into reversible sets. This is the definition that we will use in the proofs.

A sequence of incomparable pairs of with is said to be an alternating cycle of size if in for all (cyclically, so in is required). (We remark that possibly for some ’s.) Observe that if is a alternating cycle in , then this set of incomparable pairs cannot be reversed by a linear extension of . Indeed, otherwise we would have in for each cyclically, which cannot hold. Hence, alternating cycles are not reversible. The converse is also true, as is well known: A set of incomparable pairs of a poset is reversible if and only if contains no alternating cycles.

We may now turn to the proof of Theorem 2.

Proof of Theorem 2.

Let be a linear order on the elements of such that for each . Here and in the rest of the proof, weak reachability is to be interpreted w.r.t. the cover graph of and the ordering .

First, we greedily color the elements of using the ordering from left to right. When element is about to be colored, we give the smallest color in that is not used for elements of . Let and . By Observation 3, either or . Therefore,

 ϕ(y)≠ϕ(z) for every x∈P and y,z∈WReachπh−1[x] with y≠z. (1)

In the proof, we will focus on elements in that are weakly reachable from via covering chains that either start at and end in , or the other way round. This leads us to introduce the weakly reachable upset and the weakly reachable downset of :

 WU[x]:={y∈U(x):∃ % covering chain Q from x to y such that left(Q)=y}, WD[x]:={y∈D(x):∃ % covering chain Q from y to x such that left(Q)=y}.

Then . If is a set of elements of , we write for the set of colors . Given an element of and a color , by (1) there is a unique element in with color ; let us denote it by . Similarly, given , we let denote the unique element in with color .

For each , define the signature of , where

 A=ϕ(WU[x]), B=A∩ϕ(WD[y]), C={i∈B∣wui(x)<πwdi(y)}.

As , the number of possible signatures is at most . It remains to show that the set of incomparable pairs with a given signature is reversible. This will show that has dimension at most , as desired.

Arguing by contradiction, suppose that there is a signature such that the set of incomparable pairs with signature is not reversible. Then these incomparable pairs contain an alternating cycle .

For each , consider all covering chains witnessing the comparability in (indices are taken cyclically) and choose one such covering chain such that is as far to the left as possible w.r.t. . Without loss of generality we may assume that is leftmost w.r.t.  among the ’s.

Let . Clearly, in , and . Thus, and , and hence . It follows that and are both defined for each .

First suppose that . In particular, . Thus

 wut(x2)<πwdt(y2)=q1⩽πq2=left(Q2).

Since is -weakly reachable from , is -weakly reachable from , and is a path in connecting and of length at most such that , by Observation 3 one of , is weakly -reachable from the other. Since , we must have by (1). Hence, in , but , which contradicts the way (and ) was chosen.

Next, suppose that . Then, . Note that , since otherwise we would have in . Thus, , and

 wdt(y1)<πwut(x1)=q1⩽πqk=left(Qk).

Since is -weakly reachable from , is -weakly reachable from , and is a path in connecting and of length at most such that , by Observation 3 one of , is weakly -reachable from the other. Since , we must have by (1). Hence, in , but , which contradicts the way (and ) was chosen. ∎

By Zhu’s theorem, if we restrict ourselves to posets with cover graphs in a fixed class with bounded expansion, then is bounded by a function of . Thus Theorem 2 implies the theorem from [12] for classes with bounded expansion. However, the above proof is much simpler and implies better bounds on the dimension than those following from previous works (see the discussion in Section 3). We see this as a first sign that weak coloring numbers are the right tool to use in this context.

In [12], it is conjectured that bounded expansion captures exactly situations where dimension is bounded by a function of the height:

Conjecture 4 ([12]).

A monotone class of graphs has bounded expansion if and only if for every fixed , posets of height at most whose cover graphs are in have bounded dimension.

While the result of [12] (reproved above) shows the forward direction of the conjecture, the backward direction remains surprisingly (and frustratingly) open.

By contrast, showing the backward direction of Theorem 1 for nowhere dense classes is a straightforward matter, as we now explain. We prove the contrapositive. Thus let be a monotone graph class which is not nowhere dense (such a class is said to be somewhere dense). Our aim is to prove that there exist and such that there are -element posets of height at most with dimension whose cover graphs are in .

Since is somewhere dense, there exists an integer (depending on ) such that for every there is a graph containing an -subdivision of as a subgraph. (An -subdivision of a graph is a subdivision such that each edge is subdivided at most times.) Since is closed under taking subgraphs, this means that for every , the class contains a graph that is an -subdivision of the cover graph of the standard example . Notice that has at most vertices. Now it is easy to see that is also the cover graph of a poset of height at most containing as an induced subposet (simply perform the edge subdivisions on the diagram of in the obvious way). Let be the number of elements of . The poset has dimension at least , and thus its dimension is since . Hence, we obtain the desired conclusion with and . This completes the proof of the backward direction of Theorem 1.

The non-trivial part of Theorem 1 is that -element posets of bounded height with cover graphs in a nowhere dense class have dimension for all . To prove this, we use the following characterization of nowhere dense classes in terms of weak coloring numbers [22]: A class is nowhere dense if and only if for every and every , every -vertex graph in the class has weak -coloring number .

We remark that it is a common feature of several characterizations in the literature that bounded expansion and nowhere dense classes can be characterized using the same graph invariants, but requiring and bounds on the invariants respectively. Thus, it is natural to conjecture the statement of Theorem 1, and indeed it appears as a conjecture in [12]. (We note that it was originally Dan Kráľ who suggested to the first author to try and show Theorem 1 right after the result in [12] was obtained.)

The bound in Theorem 2 unfortunately falls short of implying the forward direction of Theorem 1. Indeed, if the cover graph has vertices and belongs to a nowhere dense class, we only know that for every . Thus from the theorem we only deduce that for every , which is a vacuous statement since always holds.

In order to address this shortcoming, we developed a second upper bound on the dimension of a height- poset in terms of the weak -coloring number of its cover graph (for some function ) and another invariant of . This extra invariant is the smallest integer such that does not contain an -subdivision of as a subgraph, for some function . The key aspect of our bound is that, for fixed and , it depends polynomially on the weak -coloring number that is being considered. Its precise statement is as follows. (Let us remark that the particular values and used in the theorem are not important for our purposes, any functions and would have been good.)

Theorem 5.

There exists a function such that for every and , every poset of height at most whose cover graph contains no -subdivision of as a subgraph satisfies

 dim(P)⩽(3c)f(h,t),

where .

Recall that for every nowhere dense graph class and every , there exists such that no graph in contains an -subdivision of as a subgraph. Hence, Theorem 5 implies the following corollary.

Corollary 6.

For every nowhere dense class of graphs , there exists a function such that every poset of height at most whose cover graph is in satisfies

 dim(P)⩽(3c)g(h),

where .

For every integer and real number , this in turn gives a bound of on the dimension of -element posets of height at most whose cover graphs are in . Indeed, if we take , then by the aforementioned characterization of nowhere dense classes [22], and hence by the corollary. Therefore, this establishes the forward direction of Theorem 1.

Let us also point out that Corollary 6 provides another proof of the theorem from [12] for classes with bounded expansion, since is bounded by a function of only when has bounded expansion. However, the proof is more involved than that of Theorem 2 and the resulting bound on the dimension is typically larger. Indeed, the bound in Theorem 5 becomes interesting when the weak coloring number under consideration grows with the number of vertices.

Our proof of Theorem 5 takes its roots in the alternative proof due to Micek and Wiechert [17] of Walczak’s theorem [28], that bounded-height posets whose cover graphs exclude as a topological minor have bounded dimension. This proof is essentially an iterative algorithm which, if the dimension is large enough (as a function of the height), explicitly builds a subdivision of , one branch vertex at a time. This is very similar in appearance to what we would like to show, namely that if the dimension is too big, then the cover graph contains a subdivision of where each edge is subdivided a bounded number of times (by a function of the height). The heart of our proof is a new technique based on weak coloring numbers, Lemma 7, which we use to bound the number of subdivision vertices.

The paper is organized as follows. We prove Theorem 5 in Section 2. Next, we discuss in Section 3 improved bounds implied by Theorem 2 for special cases that were studied in the literature, such as for posets with planar cover graphs and posets with cover graphs of bounded treewidth. Finally, we close the paper in Section 4 with a couple open problems.

2. Nowhere Dense Classes

As discussed in the introduction, the forward direction of Theorem 1 follows from Theorem 5 combined with Zhu’s characterization of nowhere dense classes. In this section we prove Theorem 5. We begin with our key lemma.

Lemma 7.

Let be a poset of height with cover graph , let , and let . Then there exists an element such that the set satisfies

 dim(I′)⩾dim(I)/c−2.
Proof.

Fix a linear order of the vertices of witnessing . Here and in the rest of the proof, weak reachability is to be interpreted w.r.t. the cover graph and the ordering .

Let be a greedy vertex coloring of obtained by considering the vertices one by one according to , and assigning to each vertex a color different from all the colors used on vertices in . Note that for every two vertices , we know from Observation 3 that one of is weakly -reachable from the other, and thus .

For each , let denote the color of . Observe that . Given a color , let denote the unique element of colored if there is one, and leave undefined otherwise. Observe that . In particular, in .

Let with in . We claim that . Indeed, and are at distance at most in , and the element is weakly -reachable from and is weakly -reachable from . Hence, one of is -reachable from the other by Observation 3, and .

Define the signature of a pair to be the pair , where

 α(x,y):=⎧⎪⎨⎪⎩1if wτ(y)(x)=wτ(y)(y)2if wτ(y)(x)<πwτ(y)(y)3if wτ(y)(x)>πwτ(y)(y) or wτ(y)(x) is not defined.

For each color and value , let be the set of incomparable pairs such that . Note that the sets form a partition of .

Claim 8.

For each color , the sets and are reversible.

Proof.

Let . Arguing by contradiction, suppose that is not reversible, and let denote an alternating cycle. Since in , we have that . Since , it follows that is defined.

Since for every we have in (cyclically), we obtain that . However, by our signature function this implies for all if , or for all if , which cannot hold cyclically. ∎

Since

 I=⋃τ∈[c],α∈[3]Jτ,α

the previous claims imply that

 dim(I) ⩽∑τ∈[c]dim(Jτ,1)+∑τ∈[c]dim(Jτ,2)+∑τ∈[c]dim(Jτ,3) ⩽∑τ∈[c]dim(Jτ,1)+2c.

It follows that there exists a color such that . In the rest of the proof we focus on the set . Thus, denoting this set by , we have

 dim(Iτ)⩾dim(I)/c−2.

Given an element , we denote by the set of incomparable pairs such that . Note that the sets () partition .

.

Proof.

Let . Note that since for each . Thus it remains to show that .

For each , there exists a partition of into at most reversible sets. Let be disjoint reversible sets such that

 Iτ,p=⋃j∈[d]Ijτ,p,

some sets being possibly empty. We claim that the set is reversible for each . Arguing by contradiction, suppose that for some this set is not reversible. Then it contains an alternating cycle . Again for in (with ), we have , which by the signatures of these pairs implies that . As this holds cyclically, there is such that for every . However, this implies that is an alternating cycle in , which is a contradiction since this set is reversible by assumption.

Thus, is reversible for each . Since , it follows that , as desired. ∎

Now we can complete the proof of the lemma. Let be an element witnessing the maximum value in the right-hand side of the equation in Claim 9. Clearly, . Since

 dim(Iτ,q)=dim(Iτ)⩾dim(I)/c−2,

this completes the proof of the lemma. ∎

We are now ready to prove Theorem 5.

Proof of Theorem 5.

Let and . We prove the theorem with the following value for :

 f(h,t):=(m+hh), where m:=(t2)ht.

Let thus be a poset of height at most , let denote its cover graph, and let . We prove the contrapositive. That is, we assume that

 dim(P)>(3c)f(h,t),

and our goal is to show that contains a -subdivision of as a subgraph. For technical reasons, we will need to suppose also that . This can be assumed without loss of generality, because if not then , and hence by Theorem 2.111The reader might object that this makes the proof dependent on Theorem 2, while we claimed in the introduction that it was not. In order to address this perfectly valid point, let us mention that one could choose instead to add the extra assumption that in the statement of Theorem 5; this does not change the fact that it implies the forward direction of Theorem 1 (in combination with Zhu’s theorem). However, it seemed rather artificial to do so, since the theorem remains true without this technical assumption.

Claim 10.

There exists an antichain of size in such that, letting , we have , and

 dim({(x,y)∈I:q⩽y in P})

for every element such that there exists with in .

Proof.

We define the height vector of an antichain of size at most in to be the vector of heights of elements in ordered in non-increasing order and padded at the end with -entries so that the vector is of size exactly . Note that is the number of size- vectors with entries in ordered in non-increasing order. We enumerate these vectors in lexicographic order with numbers from to . Let denote the index of the height vector of in this enumeration. Notice that .

To prove the lemma, choose an antichain with such that

 dim(I)>d:=(3c)f(h,t)−value(S),

where , and with maximum. Note that is well defined as is a candidate. Note also that . We will show that and satisfy the lemma.

First assume that is such that there exists with in and , where . Let . Observe that and , since and the height of is strictly larger than the heights of all the elements in . Moreover,

 dim(I′)⩾dim(I)/3c>d/3c=(3c)f(h,t)−value(S)−1⩾(3c)f(h,t)−value(S′),

showing that was a better choice than , a contradiction. Hence, there is no such element , and it only remains to show that . Arguing by contradiction, suppose .

Consider the poset , and let . First, we claim that is reversible in . Arguing by contradiction, suppose that this set contains an alternating cycle . Since is an induced subposet of , for each , at least one and must be in (otherwise, would be an incomparable pair of ). We cannot have , because otherwise in for some , and since in this would contradict the fact that and are incomparable. Thus, . However, since in (taking indices cyclically), it follows that , a contradiction. Hence, is reversible, as claimed. It follows

 dim(IQ)⩾dim(I)−1. (2)

Applying Lemma 7 on poset and set , we obtain an element such that . (The subscript indicates that dimension is computed w.r.t. .) Here we use that has height at most that of , and thus at most , and that the cover graph of is an (induced) subgraph of (since is an upset of ), and thus . It only remains to point out that because is an induced subposet of (that is, a subset of is an alternating cycle in if and only if it is one in ), and similarly . Putting everything together, we obtain

 dim({(x,y)∈IQ:q⩽y in P}) ⩾dim(IQ)/c−2 ⩾(dim(I)−1)/c−2 ⩾dim(I)/c−3.

Now, let and . Observe that and . Moreover,

 dim(I′) ⩾dim({(x,y)∈IQ:q⩽y in P}) ⩾dim(I)/c−3 >(3c)f(h,t)−value(S)/c−3 ⩾3⋅(3c)f(h,t)−value(S)−1−3 ⩾(3c)f(h,t)−value(S′).

(For the last inequality we use that .) This shows that is a better choice than , a contradiction. ∎

Let denote an antichain given by Claim 10, and let denote the corresponding set of incomparable pairs. The next claim will be used to build the desired subdivision of , the set will be the set of branch vertices.

Claim 11.

There exist disjoint sets and such that

1. and ,

2. for all , there is such that covers in and .

Proof.

Choose disjoint sets and satisfying 2 and

 |V|=j and |R|⩾mh−j=(t2)ht−j,

with as large as possible. Note that and is a candidate, hence this choice is possible. We claim that , which implies the lemma. Arguing by contradiction, assume .

For every , there is such that in by 2. Hence, by Claim 10

 dim({(x,y)∈I:y∉⋃v∈VU(v)}) ⩾dim(I)−∑v∈Vdim({(x,y)∈I:v⩽y%in$P$}) >dim(I)−|V|⋅dim(I)/(3c) ⩾2(1−t/(3c)) >1.

(This is the place in the proof where we use our assumption that .) It follows that the left-hand side is at least . Therefore, is not empty. (Recall that the dimension of an empty set of incomparable pairs is .) Choose some element in this set.

Now, starting from the element , we go down along cover relations in the poset . Initially we set , and as long as there is a element such that covers in and

 |D(x)∩R|>|D(v)∩R|/mh−(j+1),we update v:=x. (3)

Note that the process must stop as the height of is decreasing in every move. We claim that never goes down to a minimal element nor to an element in . Indeed, if in the above procedure we are considering an element covered by , then after at most steps we are done, and hence

 |D(v)∩R|>|D(y)∩R|m(h−2)h−(j+1)=|R|m(h−2)h−(j+1)⩾mh−j−(h−2)h−(j+1)⩾mh−(j+1).

(Note that since in for all , by our choice of .) Now, if is a minimal element or , then (note that when because and is an antichain). Hence, the inequality of (3) cannot hold strictly. Therefore, at the end of the process is not a minimal element of nor is included in , as claimed.

Consider now the set consisting of all elements that are covered by in . Since , we have