Source
Stanley (2006), “An Introduction to Hyperplane Arrangements”, PCMI lecture notes (printed 2007), LIT-355, read in full in NOTE-313 together with the author’s errata of 2020.
What was actually shown
The objects. An arrangement is a finite set of affine hyperplanes in ℝⁿ. Its regions are the connected components of the complement; each is open and convex with a polyhedral closure (pp. 3–4). Its intersection poset L(A) is the set of non-empty intersections ordered by reverse inclusion, graded by codimension (Def. 1.1, Prop. 1.1). Its characteristic polynomial is χ_A(t) = Σ_{x∈L(A)} μ(x) t^{dim x}, with μ the Möbius function (Def. 1.3).
Zaslavsky’s theorem (Thm 2.5). r(A) = (−1)ⁿ χ_A(−1) and b(A) = (−1)^{rank A} χ_A(1). The first proof is the recurrence r(A) = r(A′) + r(A″) under deletion and restriction (Lemma 2.1, with the errata’s corrected Lemma 2.2). The second uses Euler characteristic and Möbius inversion.
What it implies (Cor. 2.1). r(A) and b(A) depend only on L(A). Figure 2 gives two arrangements with the same L(A) and different face structure.
All faces (Thm 2.6). f_k(A) = Σ_{x≤y in L(A), dim x = k} |μ(x, y)|.
Checks. m lines in general position in ℝ² give C(m,2) + m + 1 regions (Ex. 1.2). The braid arrangement x_i = x_j gives n! (Ex. 1.3). A graphical arrangement’s χ is the chromatic polynomial, so its regions number the acyclic orientations (Thm 2.7, Cor. 2.3). The Shi arrangement has (n + 1)^{n−1} regions (Thm 5.16, Cor. 5.11).
Separating hyperplanes (§6.1, p. 89). d(R, R′), the number of hyperplanes separating two regions, is a metric on regions, said to be “easily seen”.
What could have come out otherwise. Two arrangements with the same intersection pattern but differently shaped regions could have had different numbers of regions. They cannot.
What this does not say
- That regions are topes. The notes never use oriented-matroid language. That a region is determined by its side of each hyperplane follows from d being a metric, which is NOTE-313’s reading, not the notes’ statement (C6). The theory of topes and covectors needs an oriented-matroid source the record does not hold.
- That anything depending on region shape can be read from L(A). Only counts can. A claim about “concept regions” that depends on their shape cannot be checked from the intersection lattice.
- That the bounded-region formula is fully proved here. The recurrence proof leaves two “[why?]” steps, and the topological proof omits ψ(Γ) = 1 (p. 21; C3).
- What Zaslavsky’s own texts contain. The notes attribute Thm 2.5 to him but do not cite LIT-332 or LIT-359.
- That identifying ultrafilters of a Boolean algebra with regions is done. That is the owner’s own step (NOTE-272, NOTE-290).
Connections
- THEORY-020: Stone’s representation gives a Boolean algebra’s points as ultrafilters. Matching them to the regions of a generally non-orthogonal arrangement is the step the owner’s map row 8 needs, and neither account supplies it.