Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability
Combinatorics is a book whose main theme is the study of subsets of a finite set. It gives a thorough grounding in the theories of set systems and hypergraphs, while providing an introduction to matroids, designs, combinatorial probability and Ramsey theory for infinite sets. The gems of the theory are emphasized: beautiful results with elegant proofs. The book developed from a course at Louisiana State University and combines a careful presentation with the informal style of those lectures. It should be an ideal text for senior undergraduates and beginning graduates.
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A C P(X A C X A U B apply induction assertion assume best possible Bollobas bound colex order colouring completely Ramsey Corollary Daykin decreasing Deduce define disjoint distributive lattice elements equality iff Erdos ex(n Exercises Frankl Furthermore Hadamard matrix Hence holds for smaller hypergraph implies independent transversal induction hypothesis infinite subsets integer intersecting family intersecting r-graph Katona Kleitman Kruskal-Katona theorem least Lemma LYM inequality matroid maximal chains maximal intersecting family monotone increasing monotone increasing property natural number non-empty Note number of edges open set orthogonal pairs partially ordered set poset problem proof of Theorem property Q r-graph G r-graph of order r-sets Ramsey theory random graph rank function real numbers result Rij(A s)-saturated satisfying sequence set system Show Sperner family Sperner system Sperner's theorem subgraph Suppose symmetric chains symmetric difference threshold function topology triples trivial vectors vertex set vertices Y C P