Topic
Infinitary combinatorics
About: Infinitary combinatorics is a research topic. Over the lifetime, 280 publications have been published within this topic receiving 9366 citations.
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Book•
02 Nov 2011
TL;DR: The Foundations of Set Theory and Infinitary Combinatorics are presented, followed by a discussion of easy Consistency Proofs and Defining Definability.
Abstract: The Foundations of Set Theory. Infinitary Combinatorics. The Well-Founded Sets. Easy Consistency Proofs. Defining Definability. The Constructible Sets. Forcing. Iterated Forcing. Bibliography. Indexes.
1,506 citations
Book•
25 Dec 1995
TL;DR: The history of combinatorics can be traced back to the work of A.C. Biggs et al. as discussed by the authors, who introduced the concept of graph embeddings.
Abstract: Part 1 Structures: graphs - basic graph theory - paths and circuits, J.A. Bondy, connectivity and network flows, A. Frank, matchings and extensions, W.R. Pulleyblank, colouring, stable sets and perfect graphs, B. Toft, embeddings and minors, C. Thomassen, random graphs, M. Karonski finite sets and relations - hypergraphs, P. Duchet, partially ordered sets, W.T. Trotter matroids - matroids - fundamental concepts, D.J.A. Welsh, matroid minors, P.D. Seymour, matroid optimization and algorithms, R.E. Bixby and W.H. Cunningham symmetric structures - permutation groups, P.J. Cameron, finite geometries, P.J. Cameron, block designs, A.E. Brouwer, association schemes, A.E. Brouwer and W. Haemers, codes, J.H. van Lint combinatorial structures in geometry and number theory - extremal problems in combinatorial geometry, P. Erdos and G. Purdy, convex polytopes and related complexes, V. Klee and P. Kleinschmidt, point lattices, J.C. Lagarias, combinatorial number theory, C. Pomerance and A. Sarkozy. Part 2 Aspects: algebraic enumeration, I.M. Gessel and R.P. Stanley asymptotic enumeration methods, A.M. Odlyzko extremal graph theory, B. Bollobas extremal set systems, P. Frankl Ramsey theory, J. Nesetril discrepancy theory, J. Beck and V.T. Sos automorphism groups, isomorphism, reconstruction, L. Babai optimization, M. Grotschel and L. Lovasz computational complexity, D.B. Shmoys and E. Tardos. Part 3 Methods: polyhedral combinatorics, A. Schrijver tools from linear algebra, C.D. Godsil tools from higher algebra, N. Alon probabilistic methods, J. Spencer topological methods, A. Bjorner. Part 4 Applications: combinatorics in operations research, A. Kolen and J.K. Lenstra combinatorics in electrical engineering and statics, A. Recski combinatorics in statistical mechanics, C.D. Godsil et al combinatorics in chemistry, D.H. Rouvray applications of combinatorics to molecular biology, M.S. Waterman combinatorics in computer science, L. Lovasz et al combinatorics in pure mathematics, L. Lovasz et al. Part 5 Horizons: infinite combinatorics, A. Hajnal combinatorial games, R.K. Guy the history of combinatorics, N.L. Biggs et al.
1,143 citations
TL;DR: A recent survey of log-concave and unimodal combinatorial properties can be found in this paper, where a variety of methods for showing that a sequence is either logarithmically concave or nonnegative are presented.
Abstract: A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an$ has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.
784 citations
Book•
06 Oct 1994
TL;DR: In this article, the authors present a textbook aimed at second-year undergraduates to beginning graduates, which stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that constructive or algorithmic proof is more valuable than an existence proof.
Abstract: Combinatorics is a subject of increasing importance, owing to its links with computer science, statistics and algebra. This is a textbook aimed at second-year undergraduates to beginning graduates. It stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that a constructive or algorithmic proof is more valuable than an existence proof. The book is divided into two parts, the second at a higher level and with a wider range than the first. Historical notes are included which give a wider perspective on the subject. More advanced topics are given as projects and there are a number of exercises, some with solutions given.
471 citations
392 citations