A combinatorial geometry is a pair (X, ℱ) where X is a set of points and where ℱ is a family of subsets of X called flats such that

1. ℱ is closed under (pairwise) intersection,

2. there are no infinite chains in the poset ℱ,

3. ℱ contains the empty set, all singletons {x}, x ∈ X, and the set X itself,

4. for every flat E ∈ ℱ, E ≠ X, the flats that cover E in F partition the remaining points.

Here, F covers E in F means that E, F ∈ ℱ, but that does not hold for any G ∈ ℱ. This latter property should be familiar to the reader from geometry: the lines that contain a given point partition the remaining points; the planes that contain a given line partition the remaining points.

A trivial example of a geometry consists of a finite set X and all subsets of X as the flats. This is the Boolean algebra on X.

We remark that (1) and (2) imply that ℱ is closed under arbitrary intersection.

Example 23.1. Every linear space (as introduced in Chapter 19) gives us a combinatorial geometry on its point set X when we take as flats φ, all singletons {{x} : x ∈ X}, all lines, and X itself. The fact that the lines on a given point partition the remaining points is another way of saying that two points determine a unique line.

One of the most popular upper level mathematics courses taught at Caltech for very many years was H. J. Ryser's course Combinatorial Analysis, Math 121. One of Ryser's main goals was to show elegance and simplicity. Furthermore, in this course that he taught so well, he sought to demonstrate coherence of the subject of combinatorics. We dedicate this book to the memory of Herb Ryser, our friend whom we admired and from whom we learned much.

Work on the present book was started during the academic year 1988–89 when the two authors taught the course Math 121 together. Our aim was not only to continue in the style of Ryser by showing many links between areas of combinatorics that seem unrelated, but also to try to more-or-less survey the subject. We had in mind that after a course like this, students who subsequently attend a conference on “Combinatorics” would hear no talks where they are completely lost because of unfamiliarity with the topic. Well, at least they should have heard many of the words before. We strongly believe that a student studying combinatorics should see as many of its branches as possible.

Of course, none of the chapters could possibly give a complete treatment of the subject indicated in their titles. Instead, we cover some highlights—but we insist on doing something substantial or nontrivial with each topic. It is our opinion that a good way to learn combinatorics is to see subjects repeated at intervals.

A partially ordered set (also poset) is a set S with a binary relation ≤ (sometimes ⊆ is used) such that:

1. (i) a ≤ a for all a ∈ S (reflexivity),

2. (ii) if a ≤ b and b ≤ a then a ≤ c (transitivity),

3. (iii) if a ≤ b and b ≤ a then a = b (antisymmetry).

If for any a and b in S, either a ≤ b or b ≤ a, then the partial order is called a total order, or a linear order. If a ≤ b and a ≠ b, then we also write a < b. Examples of posets include the integers with the usual order or the subsets of a set, ordered by inclusion. If a subset of S is totally ordered, it is called a chain. An antichain is a set of elements that are pairwise incomparable.

The following theorem is due to R. Dilworth (1950). This proof is due to H. Tverberg (1967).

Theorem 6.1.Let P be a partially ordered finite set. The minimum number m of disjoint chains which together contain all elements of P is equal to the maximum number M of elements in an antichain of P.

Proof: (i) It is trivial that m ≥ M.

(ii) We use induction on |P|. If |P| = 0, there is nothing to prove. Let C be a maximal chain in P. If every antichain in P\C contains at most M – 1 elements, we are done. So assume that {a1,…, aM} is an antichain in P\C.

We first give two different formulations of a theorem known as P. Hall's marriage theorem. We give a constructive proof and an enumerative one. If A is a subset of the vertices of a graph, then denote by г(A) the set ∪a∈Aг(a). Consider a bipartite graph G with vertex set X ∪ Y (every edge has one endpoint in X and one in Y). A matching in G is a subset E1 of the edge set such that no vertex is incident with more than one edge in E1. A complete matching from X to Y is a matching such that every vertex in X is incident with an edge in E1. If the vertices of X and Y are thought of as boys and girls, respectively, or vice versa, and an edge is present when the persons corresponding to its ends have amicable feelings towards one another, then a complete matching represents a possible assignment of marriage partners to the persons in X.

Theorem 5.1.A necessary and sufficient condition for there to be a complete matching from X to Y in G is that |г(A) > |A| for every A ⊆ X.

Proof: (i) It is obvious that the condition is necessary.

(ii) Assume that |г(A)| ≥ |A| for every A ⊆ X. Let |X| = n, m < n, and suppose we have a matching M with m edges. We shall show that a larger matching exists.