The Elser nuclei sum revisited

Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.

In Elser (1984), Veit Elser studied the probabilities of clusters forming when n points are sampled randomly in a d-dimensional volume. In the process, he found a purely graph-theoretical lemma (Elser, 1984, Lemma 1), which served a crucial role in his work. For decades, the lemma stayed hidden from the eyes of combinatorialists in a physics journal, until it resurfaced in recent work  by Dorpalen-Barry, Hettle, Livingston, Martin, Nasr, Vega and Whitlatch. In this note, I will show a simpler proof of the lemma using a sign-reversing involution. The proof also suggests multiple venues of generalization that I will explore in the later sections; one extends the lemma to a statement about arbitrary antimatroids (and even a wider setting). Finally, I will strengthen the lemma to a Morse-theoretical result, stating the collapsibility of a certain simplicial complex. Some open questions will be posed.

Remark on alternative versions
The current version of this paper is written with a combinatorially experienced reader in mind. The previous arXiv version Grinberg (2021) includes more details in the proofs.

Elser's result
Let us first introduce our setting, which is slightly more general (and perhaps also simpler) than that used in Elser (1984). (In Section 4, we will move to a more general setup. ) We fix an arbitrary graph Γ with vertex set V and edge set E. Here, "graph" means "finite undirected multigraph" -i.e., it can have self-loops and parallel edges, but it has finitely many vertices and edges, and its edges are undirected.
We fix a vertex v ∈ V . If F ⊆ E, then an F -path shall mean a path of Γ such that all edges of the path belong to F .
If e ∈ E is any edge and F ⊆ E is any subset, then we say that F infects e if there exists an F -path from v to some endpoint of e. (The terminology is inspired by the idea of an infectious disease starting in the vertex v and being transmitted along edges.) (i) A subset F ⊆ E is said to be pandemic if it infects each edge e ∈ E.
(i) Note that if an edge e contains the vertex v, then any subset F of E (even the empty one) infects e, since there is a trivial (edgeless) F -path from v to v.
We note that the equality (1) can be restated as "there are equally many pandemic subsets F ⊆ E of even size and pandemic subsets F ⊆ E of odd size". Thus, in particular, the number of all pandemic subsets F of E is even (when E = ∅).
Remark 1.4. Theorem 1.2 is a bit more general than (Elser, 1984, Lemma 1). To see why, we assume that the graph Γ is connected and simple (i.e., has no self-loops and parallel edges). Then, a nucleus is defined in Elser (1984) as a subgraph N of Γ with the properties that 1. the subgraph N is connected, and 2. each edge of Γ has at least one endpoint in N .
Given a subgraph N of Γ, we let E (N ) denote the set of all edges of N . Now, (Elser, 1984, Lemma 1)

But this is equivalent to (1), because there is a bijection
{nuclei containing v} → {pandemic subsets F ⊆ E} , N → E (N ) .
We leave it to the reader to check this in detail; what needs to be checked are the following three statements: • If N is a nucleus containing v, then E (N ) is a pandemic subset of E.
• Every nucleus N containing v is uniquely determined by the set E (N ). (Indeed, since a nucleus has to be connected, each of its vertices must be an endpoint of one of its edges, unless its only vertex is v.) • If F is a pandemic subset of E, then there is a nucleus N containing v such that E (N ) = F . (Indeed, N can be defined as the subgraph of Γ whose vertices are the endpoints of all edges in F as well as the vertex v, and whose edges are the edges in F . To see that this subgraph N is connected, it suffices to argue that each of its vertices has a path to v; but this follows from the definition of "pandemic", since each vertex of N other than v belongs to at least one edge in F .) Thus, Theorem 1.2 is equivalent to (Elser, 1984, Lemma 1)  We shall first introduce some concepts and notations pertaining to arbitrary sets. They will aid us in proving Theorem 1.2, and also in generalizing it later on.
The following simple fact abstracts an idea that will be used at least twice: Proof: This is a standard argument in enumerative combinatorics (see (Sagan, 2020, (2.3)) or Benjamin and Quinn (2008) for a more general viewpoint). Here is the proof: For each F ∈ A, we have |µ (F )| = |F | ± 1 (because (2) shows that the sets F and µ (F ) differ in exactly one element) and thus (−1) |µ(F )| + (−1) |F | = 0. This shows, in particular, that µ has no fixed points. Thus, the involution µ partitions the set to the sum on the left hand side of (3). Hence, this sum is 0. This proves Lemma 2.5.

Shades
Next, we shall introduce the notion of a shade; this will be crucial to proving and generalizing Theorem 1.2.
Definition 2.6. Let F be a subset of E. Then, we define a subset Shade F of E by We refer to Shade F as the shade of F . The major property of shades that we will need is the following: and Proof: There is no F -path from v to any endpoint of u (since u / ∈ Shade F ). Hence, any (F ∪ {u})-path that starts at v must be an F -path (as it would otherwise use the edge u and thus contain an F -path from v to some endpoint of u). This entails Shade (F ∪ {u}) ⊆ Shade F . Combined with the opposite inclusion (which follows from Lemma 2.8), this yields (5).

A slightly more general claim
Lemma 2.9 might not look very powerful, but it contains all we need to prove Theorem 1.2. Better yet, we shall prove the following slightly more general version of Theorem 1.2: Theorem 2.10. Let G be any subset of E. Assume that E = ∅. Then, We will soon prove Theorem 2.10 and explain how Theorem 1.2 follows from it. First, however, let us give an equivalent (but slightly easier to prove) version of Theorem 2.10: Theorem 2.11. Let G be any subset of E. Then, Proof: Thus, A is a subset of P (E), and each F ∈ A satisfies G ⊆ Shade F . We equip the finite set E with a total order (chosen arbitrarily, but fixed henceforth). If F ∈ A, then we define ε (F ) to be the smallest edge e ∈ G \ Shade F . (Such an edge exists, since F ∈ A entails G ⊆ Shade F and thus G \ Shade F = ∅.) For any F ∈ A, we have ε (F ) / ∈ Shade F (by the definition of ε (F )). Thus, any F ∈ A satisfies Shade (F ∪ {ε (F )}) = Shade F (by (5)) and Shade (F \ {ε (F )}) = Shade F (by (6)). In other words, if we replace a set F ∈ A by F ∪ {ε (F )} or F \ {ε (F )}, then Shade F does not change. Hence, ε (F ) does not change either (since ε (F ) depends only on Shade F , but not on F itself). Furthermore, the resulting set (F ∪ {ε (F )} or F \ {ε (F )}) still belongs to A (since Shade F has not changed). Thus, we can define a map Clearly, this map µ is an involution (since we have shown that ε (F ) does not change when we replace F by µ (F )). Moreover, this map µ is a complete matching (since each F ∈ A satisfies F ≺ µ (F ) if ε (F ) / ∈ F , and satisfies µ (F ) ≺ F otherwise). Hence, Lemma 2.5 yields F ∈A (−1) |F | = 0. In view of how we defined A, this is equivalent to (7). Thus, (7) is proven.
In order to derive Theorem 2.10 from Theorem 2.11, we need the following innocent lemma: Lemma 2.12. Let U be a finite set with U = ∅. Then, Proof: This is an easy (and well-known) consequence of Lemma 2.5. It also follows from the well-known binomial identity n k=0 (−1) k n k = 0 that holds for any integer n > 0.
We can now easily derive Theorem 2.10 from Theorem 2.11: Proof of Theorem 2.10: We have However, Lemma 2.12 (applied to U = E) shows that the left hand side of this equality is 0. Thus, so is the right hand side. This proves Theorem 2.10.

Proving Theorem 1.2
Proof of Theorem 1.2: Theorem 1.2 follows by applying Theorem 2.10 to G = E (since a subset F of E satisfies E ⊆ Shade F if and only if it is pandemic).

Vertex infection and other variants
In our study of graphs so far, we have barely ever mentioned vertices (even though they are, of course, implicit in the notion of a path). Even though the infection is spread from vertex to vertex, our sets so far have infected edges (not vertices). One might thus wonder if there is also a vertex counterpart of Theorem 1.2. So let us define analogues of our notions for vertices: If F ⊆ V , then an F -vertex-path shall mean a path of Γ such that all vertices of the path except (possibly) for its two endpoints belong to F . (Thus, if a path has only one edge or none, then it automatically is an F -vertex-path.) If w ∈ V \ {v} is any vertex and F ⊆ V \ {v} is any subset, then we say that F vertex-infects w if there exists an F -vertex-path from v to w. (This is always true when w is v or a neighbor of v.) Example 3.1. Let Γ be as in Example 1.3. Then, the path v We now have the following analogue of Theorem 1.2: Proof: With a few easy modifications, our above proof of Theorem 1.2 can be repurposed as a proof of Theorem 3.2. Most importantly, we need to replace the set E by V \ {v}, and we need to replace the words "edge", "F -path", "infects" and "pandemic" by "vertex", "F -vertex-path", "vertex-infects" and "vertex-pandemic", respectively.
Another variant of Theorem 1.2 (and Theorem 2.10 and Theorem 2.11) is obtained by replacing the undirected graph Γ with a directed graph (while, of course, replacing paths by directed paths). More generally, we can replace Γ by a "hybrid" graph with some directed and some undirected edges. (ii) No changes are required to the above proofs. Yet another variation can be obtained by replacing "endpoint" by "source" (for directed edges). We cannot, however, replace "endpoint" by "target".

An abstract perspective
Seeing how little graph theory we have used in proving Theorem 1.2, and how easily the same argument adapted to Theorem 3.2, we get the impression that there might be some general theory lurking behind it. What follows is an attempt at building this theory.
Most proofs in this section are omitted; some are outlined. In fact, they are all sufficiently simple and straightforward that the reader should have little trouble filling them in; alternatively, almost all of them can be found in the detailed version of Grinberg (2021).
(ii) We understand that a directed edge still has two endpoints: its source and its target.

Shade maps
Let P (E) denote the power set of E. In Definition 2.6, we have encoded the "infects" relation as a map Shade : P (E) → P (E) defined by Shade F = {e ∈ E | F infects e}. As we recall, Theorem 2.10 (a generalization of Theorem 1.2) states that for any G ⊆ E, under the assumption that E = ∅.
To generalize this, we forget about the graph Γ and the map Shade, and instead start with an arbitrary finite set E. (This set E corresponds to the set E in Theorem 1.2 and to the set V \ {v} in Theorem 3.2.) Let P (E) be the power set of E. Let Shade : P (E) → P (E) be an arbitrary map (meant to generalize the map Shade from the previous paragraph). We may now ask: Question 4.1. What (combinatorial) properties must Shade satisfy in order for (9) to hold for any G ⊆ E under the assumption that E = ∅ ?
A partial answer to this question can be given by analyzing our above proof of Theorem 2.10 and extracting what was used: Definition 4.2. Let E be a set. A shade map on E shall mean a map Shade : P (E) → P (E) that satisfies the following two axioms: Assume that E = ∅. Let G be any subset of E. Then, Proof: Again, the proof is analogous to our above proof of Theorem 2.10. (This time, in the proof of Lemma 2.9, the equalities (5) and (6) follow directly from Axiom 1 and Axiom 2, respectively.) How do shade maps relate to known concepts in the combinatorics of set families (such as topologies, clutters, matroids, or submodular functions)? Are they just one of these known concepts in disguise? We shall answer two versions of this question in the following subsections. Specifically: • In Subsection 4.3, we will show that inclusion-reversing shade maps on E (i.e., shade maps Shade that satisfy Shade B ⊆ Shade A whenever A ⊆ B) are in bijection with antimatroidal quasiclosure operators (a slight variant of antimatroids) on E.
• In Subsection 4.4, we will show that arbitrary shade maps are in bijection with Boolean interval partitions of P (E) (that is, set partitions of P (E) into intervals of the Boolean lattice P (E)).
Before we come to these characterizations, we shall however make a few elementary remarks on shade maps.
First, we observe that Axioms 1 and 2 in Definition 4.2 can be weakened to the following statements: Axiom 1' is weaker than Axiom 1, and likewise Axiom 2' is weaker than Axiom 2. However, Axioms 1' and 2' combined are equivalent to Axioms 1 and 2 combined: Axioms 1 and 2 can also be combined into one common axiom:

any map. Then, Shade is a shade map on E if and only if Shade satisfies Axiom 3.
We will soon see some examples. First, let us introduce two more basic concepts that will help clarify these examples: Definition 4.6. Let E be a set. Let Shade : P (E) → P (E) be any map (not necessarily a shade map).
(a) We say that Shade is inclusion-preserving if it satisfies the following property: If A and B are two subsets of E such that A ⊆ B, then Shade A ⊆ Shade B.
(b) We say that Shade is inclusion-reversing if it satisfies the following property: If A and B are two subsets of E such that A ⊆ B, then Shade B ⊆ Shade A.
For instance, the map Shade from Definition 2.6 is inclusion-preserving (because of Lemma 2.8) and is a shade map (by Lemma 2.9). The same holds for the analogue of the map Shade that uses vertexinfection instead of infection. We will soon see some inclusion-reversing shade maps, and it is not hard to construct shade maps that are neither inclusion-preserving nor inclusion-reversing.
Let us observe that there is a simple bijection between inclusion-preserving and inclusion-reversing maps, and this bijection preserves shadeness:

Some examples of shade maps
As we already mentioned, Lemma 2.9 and its analogue for vertex-infection provide two examples of inclusion-preserving shade maps Shade. An example of an inclusion-reversing shade map comes from the theory of posets: Example 4.9. Let E be a poset. For any F ⊆ E, we define Then, this map Shade : P (E) → P (E) is an inclusion-reversing shade map.
Another example of a shade map comes from discrete geometry: Example 4.10. Let A be an affine space over R. If S is a finite subset of A, then a nontrivial convex combination of S will mean a point of the form s∈S λ s s ∈ A, where the coefficients λ s are nonnegative reals smaller than 1 and satisfying s∈S λ s = 1.
Then, this map Shade : P (E) → P (E) is an inclusion-reversing shade map.
As a contrast to Example 4.10, let us mention a not-quite-example (satisfying only one of the two axioms in Theorem 4.3): Example 4.11. Let V be a vector space over R. If S is a finite subset of V , then a nontrivial conic combination of S will mean a vector of the form s∈S λ s s ∈ V , where the coefficients λ s are nonnegative reals with the property that at least two elements s ∈ S satisfy λ s > 0.
Fix a finite subset E of V . For any F ⊆ E, we define It can be shown that this map Shade : P (E) → P (E) satisfies Axiom 1 in Definition 4.2. In general, it does not satisfy Axiom 2. Thus, it is not a shade map in general.

Antimatroids and inclusion-reversing shade maps
Examples 4.9 and 4.10 are instances of a general class of examples: shade maps coming from antimatroids. Not unlike matroids, antimatroids are a combinatorial concept with many equivalent avatars (see, e.g., (Korte et al., 1991, Chapter III)). Here we shall view them through one of these avatars: that of antimatroidal quasi-closure operators (roughly equivalent to convex geometries). We begin by defining the notions we need: Definition 4.12. Let E be any set.
(a) A quasi-closure operator on E means a map τ : P (E) → P (E) with the following properties: 1. We have A ⊆ τ (A) for any A ⊆ E.
2. If A and B are two subsets of E satisfying A ⊆ B, then τ (A) ⊆ τ (B).
(b) A quasi-closure operator τ on E is said to be antimatroidal if it has the following additional property: 4. If X is a subset of E, and if y and z are two distinct elements of E \ τ (X) satisfying z ∈ τ (X ∪ {y}), then y / ∈ τ (X ∪ {z}).
(c) A closure operator on E means a quasi-closure operator τ on E that satisfies τ (∅) = ∅.
(d) If τ is an antimatroidal closure operator on E, then the pair (E, τ ) is called a convex geometry.
Here are some examples of antimatroidal quasi-closure operators: Example 4.13. Let E be a poset. For any F ⊆ E, we define Then, τ is an antimatroidal closure operator on E. (This example is the "downset alignment" from (Edelman and Jamison, 1985, §3, Example II), and is equivalent to the "poset antimatroid" from (Korte et al., 1991, §III.2

.3).)
Example 4.14. Let E be a poset. For any F ⊆ E, we define Then, τ is an antimatroidal closure operator on E. (This example is the "order convex alignment" from (Edelman and Jamison, 1985, §3, Example II), and is the "double shelling of a poset" example from (Korte et al., 1991, §III.2.4).) Example 4.15. Let A be an affine space over R. If S is a finite subset of A, then a convex combination of S will mean a point of the form s∈S λ s s ∈ A, where the coefficients λ s are nonnegative reals satisfying Then, τ is an antimatroidal closure operator on E. (This example is (Edelman and Jamison, 1985, §3, Example I); it gave the name "convex geometry" to the notion defined in Definition 4.12 (d).) Example 4.16. Let Γ be any graph with edge set E. Fix a vertex v of Γ. We say that a subset F ⊆ E blocks an edge e ∈ E if each path of Γ that contains v and e must contain at least one edge of F . (In particular, this is automatically the case when e ∈ F .) For each F ⊆ E, we define Then, τ is an antimatroidal quasi-closure operator on E. (This example is the "line-search antimatroid" from (Korte et al., 1991, §III.2.11).) If Γ is connected, then τ is actually a closure operator.
We shall be dealing with quasi-closure operators rather than closure operators most of the time. However, since the latter concept is somewhat more widespread, let us comment on the connection between the two. Roughly speaking, the relation between quasi-closure and closure operators is comparable to the relation between semigroups and monoids, or between nonunital rings and unital rings, or (perhaps the best analogue) between simplicial complexes in general and simplicial complexes without ghost vertices (i.e., simplicial complexes for which every element of the ground set is a dimension-0 face). More concretely, specifying a quasi-closure operator on a set E is tantamount to specifying a subset of E and a closure operator on this subset. To wit: Then, this map Shade : P (E) → P (E) is an inclusion-reversing shade map.
Theorem 4.18 generalizes Examples 4.9 and 4.10. Indeed, the latter two examples are obtained by applying Theorem 4.18 to the settings of Examples 4.13 and 4.15, respectively. Less directly, Lemma 2.9 and its vertex-infection analogue are particular cases of Theorem 4.18 as well (even though they involve shade maps that are inclusion-preserving rather than inclusion-reversing). Indeed, if we apply Theorem 4.18 to the setting of Example 4.16, then we obtain the claim of Lemma 2.9 with Shade F replaced by Shade (E \ F ); this is easily seen to be equivalent to Lemma 2.9 (by the duality stated in Proposition 4.7).
Our proof of Theorem 4.18 will rely on two easy lemmas: Lemma 4.19. Let E be a set. Let τ be a quasi-closure operator on E. Let X be a subset of E, and let z ∈ τ (X). Then, τ (X ∪ {z}) = τ (X).
Lemma 4.20. Let E be a set. Let τ be an antimatroidal quasi-closure operator on E. Let X be a subset of E, and let y and z be two distinct elements of E satisfying z ∈ τ (X ∪ {y}) and y ∈ τ (X ∪ {z}). Then, y ∈ τ (X).
Note that Lemma 4.20 has a converse: If τ is a quasi-closure operator on E satisfying the claim of Lemma 4.20, then τ is antimatroidal. This is easy to see but will not be used in what follows.
Proof of Theorem 4.18: We shall prove the following three statements: Our proofs of these three statements will use Property 2 in Definition 4.12 (a) (which we shall just refer to as "Property 2"). [ However, from v ∈ Shade (F \ {u}), we obtain v / ∈ τ ((F \ {u}) \ {v}) (by the definition of Shade (F \ {u})). In other words, v / ∈ τ (X) (since X = (F \ {u}) \ {v}). However, if v and u were distinct, then Lemma 4.20 (applied to y = v and z = u) would yield v ∈ τ (X) (since v ∈ τ (X ∪ {u}) and u ∈ τ (X ∪ {v})), which would contradict v / ∈ τ (X). Thus, v and u cannot be distinct. In other words, v = u. Now, We note that the quasi-closure operator τ in Theorem 4.18 can be reconstructed from the map Shade. This does not even require τ to be antimatroidal; the following holds for any quasi-closure operator: It turns out that if one applies the formula (11) to an inclusion-reversing shade map Shade, then the resulting map τ is an antimatroidal quasi-closure operator, at least when E is finite. In fact, we have the following: Proposition 4.22. Let E be a finite set. Let Shade : P (E) → P (E) be an inclusion-reversing shade map. Define a map τ : P (E) → P (E) by setting for each F ⊆ E. Then, τ is an antimatroidal quasi-closure operator on E.
The proof of this proposition rests on the following lemma: Then, each F ⊆ E satisfies Combining many of the results in this section, we obtain the following description of inclusion-reversing shade maps: Theorem 4.25 classifies inclusion-reversing shade maps in terms of antimatroidal quasi-closure operators (iii) . The latter can in turn be described in terms of antimatroidal closure operators (by Proposition 4.17), i.e., in terms of antimatroids. Thus, inclusion-reversing shade maps "boil down" to antimatroids. The same can be said of inclusion-preserving shade maps (because Proposition 4.7 establishes a bijection between them and the inclusion-reversing ones). In the next subsection, we shall classify arbitrary shade maps in terms of what we will call Boolean interval partitions.

Boolean interval partitions and arbitrary shade maps
Let us first define Boolean interval partitions: Definition 4.26. Let E be a set.
(a) If U and V are two subsets of E satisfying U ⊆ V , then [U, V ] shall denote the subset . This is the set of all subsets of E that lie between U and V (meaning that they contain U as a subset, but in turn are contained in V as subsets). (c) A Boolean interval partition of P (E) means a set of pairwise disjoint Boolean intervals of P (E) whose union is P (E).
(d) If P is a Boolean interval partition of P (E), then the elements of P (that is, the Boolean intervals that belong to P) are called the blocks of P. The former has four blocks; the latter has five.
Here are two ways to think of Boolean interval partitions of P (E): • The following is just a slick restatement of Definition 4.26 (c) using standard combinatorial lingo: A Boolean interval partition of P (E) is a set partition of the Boolean lattice P (E) into intervals.
• It is well-known that the set partitions of a given set are in a canonical bijection with the equivalence relations on this set. In light of this, the Boolean interval partitions of P (E) can be viewed as the equivalence relations on P (E) whose equivalence classes are Boolean intervals. In other words, they can be viewed as the equivalence relations ∼ on P (E) satisfying the axiom "if U, V, I ∈ P (E) satisfy U ∼ V and U ∩ V ⊆ I ⊆ U ∪ V , then U ∼ I ∼ V ". The reader can prove this alternative characterization as an easy exercise in Boolean algebra.
We shall now construct a shade map from any Boolean interval partition: Theorem 4.28. Let E be a set. Let P be a Boolean interval partition of P (E).
For any F ∈ P (E), let [α (F ) , τ (F )] denote the (unique) block of P that contains F . We define a map Shade : P (E) → P (E) by setting for any F ∈ P (E) . Then: The proof of Theorem 4.28 is rather easy. We lighten our burden somewhat with a simple lemma (which can be easily checked using Venn diagrams): Lemma 4.29. Let X, Y , Z and E be four sets such that X ⊆ Y ⊆ Z ⊆ E. Then: Proof of Theorem 4.28: (a) We shall prove the following two statements: [Proof of Statement 1: Let F ∈ P (E) and u ∈ E \ Shade F . We must prove that The definition of α (F ) and τ (F ) reveals that [α (F ) , τ (F )] is the (unique) block of P that contains F . Thus, [α (F ) , τ (F )] is a block of P and contains F .
On the other hand, the Boolean interval [α (F ) , τ (F )] contains F . In other words, is a block of P that contains F ′ (since we already know that [α (F ) , τ (F )] is a block of P).
However, the definition of α (F ′ ) and τ (F ′ ) shows that [α (F ′ ) , τ (F ′ )] is the (unique) block of P that contains F ′ . Since we know that [α (F ) , τ (F )] is a block of P that contains F ′ , we therefore conclude (because a Boolean interval [U, V ] uniquely determines both U and V ). Now, the definition of Shade yields In view of F ′ = F ∪ {u}, this rewrites as Shade (F ∪ {u}) = Shade F . This proves Statement 1.] [Proof of Statement 2: Let F ∈ P (E) and u ∈ E \ Shade F . We must prove that Shade (F \ {u}) = Shade F .
We proceed exactly as in our above proof of Statement 1 up until the point where we define F ′ . Insead of setting . From this, we can obtain Shade (F ′ ) = Shade F by the same argument that we used back in the proof of Statement 1. In view of F ′ = F \ {u}, this rewrites as Shade (F \ {u}) = Shade F . This proves Statement 2.] Now, we have proved Statements 1 and 2. Thus, the map Shade : P (E) → P (E) satisfies the two axioms in Definition 4.2. In other words, this map is a shade map. This proves Theorem 4.28 (a).
A converse to Theorem 4.28 is provided by the following theorem: Theorem 4.30. Let E be a finite set. Let Shade : P (E) → P (E) be a shade map on E. Define a map α : P (E) → P (E) by setting Define a map τ : P (E) → P (E) by setting Then: To prove this theorem, we will need the following variant of Lemma 4.23: Proof of Lemma 4.31: Induction on |B|, using Axiom 2 from Definition 2.6.
We will also use another simple set-theoretical lemma (easily checked using Venn diagrams): Lemma 4.32. Let E be a set. Let X and Y be two subsets of E. Then, Proof of Theorem 4.30: (a) Let F ∈ P (E). The definition of α yields α In other words, F ∈ [α (F ) , τ (F )]. This proves Theorem 4.30 (a).
Furthermore, the definition of α yields F ∩ Shade F = α (F ) ⊆ G. From this, straightforward settheoretical reasoning leads to Hence, Lemma 4.31 (applied to A = F and B = F \ G) yields Shade (F \ (F \ G)) = Shade F . In view of F \ (F \ G) = F ∩ G, this rewrites as Shade (F ∩ G) = Shade F . Hence, (14) rewrites as Thus, Lemma 4.23 (applied to A = F ∩ G and B = G \ F ) yields By further set-theoretical reasoning, using the equalities (14) and (15) (and nothing else specific to the sets F , G and Shade F ), we can easily obtain In view of we can rewrite these two equalities as α (F ) = α (G) and τ (F ) = τ (G). (c) Clearly, P is a set of Boolean intervals of P (E). Moreover, the Boolean intervals [α (F ) , τ (F )] ∈ P are pairwise disjoint (because if two of these intervals have some element G in common, then Theorem 4.30 (b) yields that they must both equal [α (G) , β (G)]), and their union is P (E) (since every F ∈ P (E) satisfies F ∈ [α (F ) , τ (F )] by Theorem 4.30 (a)). Thus, P is a Boolean interval partition of P (E). This proves Theorem 4.30 (c).
(d) Let F ∈ P (E). Then, Lemma 4.32 (applied to X = F and Y = Shade F ) yields However, Shade F is a subset of E; thus, This proves Theorem 4.30 (d).
Combining Theorem 4.28 with Theorem 4.30, we obtain the following:  Gordon and McMahon in (Gordon and McMahon, 1997, Theorem 2.5)?

The topological viewpoint
Now we return to the setting of Section 1. We aim to reinterpret Theorem 2.11 in the terms of combinatorial topology (specifically, finite simplicial complexes) and strengthen it. We recall the definition of a simplicial complex: (iv) Definition 5.1. Let E be a finite set. A simplicial complex on ground set E means a subset A of the power set of E with the following property: If P ∈ A and Q ⊆ P , then Q ∈ A.
Thus, in terms of posets, a simplicial complex on ground set E means a down-closed subset of the Boolean lattice on E. Note that a simplicial complex contains the empty set ∅ unless it is empty itself.
We refer to Kozlov (2020) for context and theory about simplicial complexes. We shall restrict ourselves to the few definitions relevant to what we will prove. The following is straightforward to check: Proposition 5.2. Let us use the notations from Section 1 as well as Definition 2.6. Let G be any subset of Then, A is a simplicial complex on ground set E.
Example 5.3. The following pictures illustrate this simplicial complex on an example. The left picture is a graph Γ (with the vertex labelled v playing the role of v), whereas the right picture shows the corresponding simplicial complex A for G = E (that is, the simplicial complex whose faces are the subsets of E that are not pandemic).
To state the main result of this section, we need the following definition (v) : Definition 5.4. Let E be a finite set. Let A be a simplicial complex on ground set E.
(a) A complete matching µ of A is said to be acyclic if there exists no tuple (B 1 , B 2 , . . . , B n ) of distinct sets B 1 , B 2 , . . . , B n ∈ A with the property that n ≥ 2 and that The simplicial complex A is said to be collapsible if it has an acyclic complete matching.
Note that our definition of an "acyclic complete matching" is a particular case of (Kozlov, 2020, Definition 10.7). Our notion of "collapsible" is equivalent to the classical notion of "collapsible" (even though the latter is usually defined differently) because of (Kozlov, 2020, Theorem 10.9).
We now claim: Theorem 5.5. Let us use the notations from Section 1 as well as Definition 2.6. Let G be any subset of E. Define A as in (16). Then, the simplicial complex A is collapsible.
Collapsible simplicial complexes are well-behaved in various ways -in particular, they are contractible ( (Kozlov, 2020, Corollary 9.19)), and thus have trivial homotopy and homology groups (in positive degrees). Moreover, the reduced Euler characteristic of any collapsible simplicial complex is 0 (for obvious reasons: having a complete matching suffices, even if it is not acyclic); thus, Theorem 2.11 follows from Theorem 5.5.
Our proof of Theorem 5.5 will rely on the following simple lemma (whose proof is left as an exercise): Lemma 5.6. Let X and Y be two sets, and let u ∈ X ∩ Y . If X \ {u} ≺ Y , then X = Y .
Proof of Theorem 5.5: We know from Proposition 5.2 that A is a simplicial complex. It remains to show that A is collapsible. Note that our A is precisely the set A defined in the proof of Theorem 2.11 above. We equip the finite set E with a total order (chosen arbitrarily). If F ∈ A, then we define the edge ε (F ) ∈ G \ Shade F as in the proof of Theorem 2.11. We define a complete matching µ : A → A of A as in the proof of Theorem 2.11. We shall now prove that this complete matching µ is acyclic.
Forget that we fixed (B 1 , B 2 , . . . , B n ). We thus have found a contradiction whenever (B 1 , B 2 , . . . , B n ) is a tuple of distinct sets B 1 , B 2 , . . . , B n ∈ A with the property that n ≥ 2 and that (17) and (18) and (19). Hence, there exists no such tuple. In other words, the complete matching µ is acyclic. Therefore, the simplicial complex A is collapsible (by Definition 5.4 (b)). This finishes the proof of Theorem 5.5.
The analogues of Proposition 5.2 and of Theorem 5.5 for vertex-infection (instead of usual infection) also hold (with the same proofs). More generally, Proposition 5.2 and Theorem 5.5 can be generalized to any inclusion-preserving shade map: Theorem 5.7. Let E be any set. Let Shade : P (E) → P (E) be an inclusion-preserving shade map on E. Let G be any subset of E. Let

Then:
(a) This A is a simplicial complex on ground set E.
(b) This simplicial complex A is collapsible.