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9084 results in Discrete Mathematics, Information Theory and Coding

Page 7 of 455

Oriented Matroids
  • Laura Anderson
  • Published online:
    04 June 2025
    Print publication:
    10 April 2025
  • Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.

Games of No Chance 4
  • Edited by Richard J. Nowakowski
  • Mathematical Sciences Research Institute
  • Published online:
    30 May 2025
    Print publication:
    16 April 2015
  • Combinatorial games are the strategy games that people like to play, for example chess, Hex, and Go. They differ from economic games in that there are two players who play alternately with no hidden cards and no dice. These games have a mathematical structure that allows players to analyse them in the abstract. Games of No Chance 4 contains the first comprehensive explorations of misère (last player to move loses) games, extends the theory for some classes of normal-play (last player to move wins) games and extends the analysis for some specific games. It includes a tutorial for the very successful approach to analysing misère impartial games and the first attempt at using it for misère partisan games. Hex and Go are featured, as well as new games: Toppling Dominoes and Maze. Updated versions of Unsolved Problems in Combinatorial Game Theory and the Combinatorial Games Bibliography complete the volume.

  • Tight Hamilton cycles with high discrepancy

  • Part of
  • Lior Gishboliner, Stefan Glock, Amedeo Sgueglia
  • Journal:
    Combinatorics, Probability and Computing / Volume 34 / Issue 4 / July 2025
    Published online by Cambridge University Press:
    30 May 2025, pp. 565-584
    • Article
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  • In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta (G) \ge (1/2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n/r+\Omega (n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Turán-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.

Games of No Chance 5
  • Edited by Urban Larsson
  • Mathematical Sciences Research Institute
  • Published online:
    29 May 2025
    Print publication:
    09 May 2019
  • This book surveys the state-of-the-art in the theory of combinatorial games, that is games not involving chance or hidden information. Enthusiasts will find a wide variety of exciting topics, from a trailblazing presentation of scoring to solutions of three piece ending positions of bidding chess. Theories and techniques in many subfields are covered, such as universality, Wythoff Nim variations, misère play, partizan bidding (a.k.a. Richman games), loopy games, and the algebra of placement games. Also included are an updated list of unsolved problems, extremely efficient algorithms for taking and breaking games, a historical exposition of binary numbers and games by David Singmaster, chromatic Nim variations, renormalization for combinatorial games, and a survey of temperature theory by Elwyn Berlekamp, one of the founders of the field. The volume was initiated at the Combinatorial Game Theory Workshop, January 2011, held at the Banff International Research Station.

More Games of No Chance
  • Edited by Richard Nowakowski
  • Mathematical Sciences Research Institute
  • Published online:
    29 May 2025
    Print publication:
    25 November 2002
  • This 2003 book provides an analysis of combinatorial games - games not involving chance or hidden information. It contains a fascinating collection of articles by some well-known names in the field, such as Elwyn Berlekamp and John Conway, plus other researchers in mathematics and computer science, together with some top game players. The articles run the gamut from theoretical approaches (infinite games, generalizations of game values, 2-player cellular automata, Alpha-Beta pruning under partial orders) to other games (Amazons, Chomp, Dot-and-Boxes, Go, Chess, Hex). Many of these advances reflect the interplay of the computer science and the mathematics. The book ends with a bibliography by A. Fraenkel and a list of combinatorial game theory problems by R. K. Guy. Like its predecessor, Games of No Chance, this should be on the shelf of all serious combinatorial games enthusiasts.

Thin Groups and Superstrong Approximation
  • Edited by Emmanuel Breuillard, Hee Oh
  • Mathematical Sciences Research Institute
  • Published online:
    29 May 2025
    Print publication:
    17 February 2014
  • This collection of survey and research articles focuses on recent developments concerning various quantitative aspects of 'thin groups'. There are discrete subgroups of semisimple Lie groups that are both big (i.e. Zariski dense) and small (i.e. of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading, for instance, to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap, referred to as superstrong approximation. This book provides a broad panorama of a very active field of mathematics at the boundary between geometry, dynamical systems, number theory and combinatorics. It is suitable for professional mathematicians and graduate students in mathematics interested in this fascinating area of research.

Games of No Chance 6
  • Edited by Urban Larsson
  • Published online:
    18 May 2025
    Print publication:
    05 June 2025
  • This collection of 22 research papers and state-of-the-art surveys extends the subseries 'Games of No Chance' pioneered in 1996. Survey topics include Richman bidding combinatorial games, classical subtraction games and absolute additive theory. Other topics discussed include extensions of normal play theory such as Absolute CGT and Affine normal play; additive theory; aspects of generic impartial games arising from the study of nim-values; dead-ending misère reduction theorems; Wythoff-type variations; complexity issues; and aspects of classical games including a rigorous justification of the celebrated result that king, bishop and knight can checkmate a lonely king on an arbitrarily large chessboard. The recurring list of open problems, updated and annotated, will interest all practitioners of CGT and related fields including algebra, computer science, combinatorics, number theory and classical game theory.

  • Short proof of the hypergraph container theorem

  • Part of
  • Rajko Nenadov, Huy Tuan Pham
  • Journal:
    Combinatorics, Probability and Computing / Volume 34 / Issue 5 / September 2025
    Published online by Cambridge University Press:
    16 May 2025, pp. 621-624

  • We present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.

  • Colouring random subgraphs

  • Part of
  • Boris Bukh, Michael Krivelevich, Bhargav Narayanan
  • Journal:
    Combinatorics, Probability and Computing / Volume 34 / Issue 4 / July 2025
    Published online by Cambridge University Press:
    28 April 2025, pp. 585-595

  • We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

Page 7 of 455