One square and an odd number of triangles. Van der Waerden's permanent conjecture. On a lemma of Littlewood and Offord. Cotangent and the Herglotz trick. Buffon's needle problem. Pigeon-hole and double counting. Tiling rectangles. Three famous theorems on finite sets. Shuffling cards. Lattice paths and determinants.
Cayley's formula for the number of trees. Identities versus bijections. The finite Kakeya problem. Completing Latin squares. Permanents and the power of entropy.
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The Dinitz problem. Five-coloring plane graphs. How to guard a museum. Communicating without errors. The chromatic number of Kneser graphs. Of friends and politicians. Probability makes counting sometimes easy. Anmeldung Mein Konto Merkzettel 0. Polytopes are certainly familiar objects: Prime examples are given by convex polygons 2-dimensional convex polytopes and by convex polyhedra 3-dimensional convex polytopes.
Familiar polytopes: tetrahedron and cube General polytopes are defined as finite unions of convex polytopes. Convex polytopes can, equivalently, be defined as the bounded solution sets of finite systems of linear inequalities. Conversely, every bounded such solution set is a convex polytope, and can thus be represented as the convex hull of a finite set of points. For polygons and polyhedra, we have the familiar concepts of vertices, edges, and 2-faces.
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All the faces of a polytope are themselves polytopes. The graph G P of the convex polytope P is given by the set V of vertices, and by the edge set E of 1-dimensional faces. Such a map may reverse the orientation of space, as does the reflection of P in a hyperplane, which takes P to a mirror image of P. In this situation we call x0 the center of P. References  V. Monthly, , Gesellschaft der Wissenschaften, Mathematisch-physikalische Klasse , Teubner, Leipzig Combinatorially equivalent polytopes Lines in the plane and decompositions of graphs Chapter 10 Perhaps the best-known problem on configurations of lines was raised by Sylvester in in a column of mathematical problems.
Therefore the following theorem is commonly attributed to Sylvester and Gallai. In any configuration of n points in the plane, not all on a line, there is a line which contains exactly two of the points. Let P be the given set of points and consider the set L of all lines which pass through at least two points of P. The figure on the right shows the configuration. Do we really need these properties beyond the usual incidence axioms of points and lines?
Well, some additional condition is required, as the famous Fano plane depicted in the margin demonstrates. Any two points determine a unique line, so the incidence axioms are satisfied, but there is no 2-point line.
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We will discuss the more general result in a moment. Then the set L of lines passing through at least two points contains at least n lines. P P3 P Now we proceed by induction on n. The following proof, variously attributed to Motzkin or Conway, is almost one-line and truly inspired. Let B be the incidence matrix of X; A1 ,. Consider the product BB T.
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Let us go a little beyond and turn to graph theory. We refer to the review of basic graph concepts in the appendix to this chapter. Indeed, let X correspond to the vertex set of Kn and the sets Ai to the vertex sets of the cliques, then the statements are identical. Our next task is to decompose Kn into complete bipartite graphs such that again every edge is in exactly one of these graphs.
There is an easy way to do this. First take the complete bipartite graph joining 1 to all other vertices. Next join 2 to 3,. Can we do better, that is, use fewer graphs? No, as the following result of Ron Graham and Henry O. Pollak says: Theorem 4. If Kn is decomposed into complete bipartite subgraphs H1 ,. All of them use linear algebra in one way or another. Of the various more or less equivalent ideas let us look at the proof due to Tverberg, which may be the most transparent. To every vertex i we associate a variable xi.
Since H1 ,. It has one loop, one double edge and one triple edge.
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Graphs are among the most basic of all mathematical structures. Correspondingly, they have many different versions, representations, and incarnations. We consider only finite graphs, where V and E are finite. Usually, we deal with simple graphs: Then we do not admit loops, i. Vertices of a graph are called adjacent or neighbors if they are the endvertices of an edge. A vertex and an edge are called incident if the edge has the vertex as an endvertex.
Here is a little picture gallery of important simple graphs: K2 K3 K4 K K2,3 K3, It is a major unsolved problem whether there is an efficient test to decide whether two given graphs are isomorphic. This notion of isomorphism allows us to talk about the complete graph K5 on 5 vertices, etc. Many notions about graphs are quite intuitive: for example, a graph G is connected if every two distinct vertices are connected by a path in G, or equivalently, if G cannot be split into two nonempty subgraphs whose vertex sets are disjoint.
Any graph decomposes into its connected components. We end this survey of basic graph concepts with a few more pieces of terminology: A clique in G is a complete subgraph. An independent set in G is an induced subgraph without edges, that is, a subset of the vertex set such that no two vertices are connected by an edge of G. A graph is a forest if it does not contain any cycles.
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Graph Theory 6 , But of course many of these lines may be parallel, and thus determine the same slope. A little experimentation for small n will probably lead you to a sequence such as the two depicted here. However, the configurations that we have drawn above are by far not the only ones.