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9031 results in Discrete Mathematics, Information Theory and Coding

Page 5 of 452

  • 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
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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 $k/2 - o(k)$.

  • 7 - The Universality Theorem

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 214-241

  • Summary

    A discussion of realization spaces, including an example of an oriented matroid with disconnected extension space, is provided. In the later part, a proof of the Universality Theorem and a discussion of some of its consequences follows.

  • 2 - Oriented Matroids

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 43-91

  • Summary

    The chapters addresses the various axiomatizations and the equivalences between them and presents an introduction to the Plucker relations. The chapter finishes with some discussion of nonrealizable oriented matroids and the impossibility of an excluded minor characterization.

  • 1 - Realizable Oriented Matroids

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 1-42
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  • Summary

    The geometric motivation for the theory is combinatorial data associated with matrices, vector arrangements, hyperplane arrangements, and subspaces of real vector spaces. Interpretations of this data are given in terms of linear algebra, discrete geometry, and the Plucker embedding of the Grassmannian. Elementary proofs of cryptomorphisms for realizable oriented matroids are provided. The chapter finishes with an application of Gale Diagrams.

  • 10 - Spaces of Oriented Matroids

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 296-307

  • Summary

    The chapters provides a survey on the topology of various posets of oriented matroids analogous to various topological spaces, including extension spaces, combinatorial Grassmannians, and combinatorial flag spaces. A general framework for interpreting maps from spaces to posets is laid down, by way of McCord’s Theorem and the Semi-algebraic Triangulation Theorem. The chapter includes a discussion of the (now-disproved) extension space conjecture and of the various results on the topology of the MacPhersonian.

  • 8 - Oriented Matroid Polytopes

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 242-257

  • Summary

    Introduction to the oriented matroid abstraction of convex polytopes is the goal of this chapter. The chapter shows that the existence of a polar of an oriented matroid polytope is closely related to the existence of an adjoint. The Lawrence construction is introduced and used to produce interesting examples.

  • 6 - Single-Element Extensions

  • Laura Anderson, Binghamton University, New York
  • Book:
    Oriented Matroids
    Published online:
    04 June 2025
    Print publication:
    10 April 2025, pp 158-213

  • Summary

    The chapter begins with localizations, including a topological proof of Las Vergnas’s characterization of localizations. Adjoints and their relationship to extensions are discussed. The final part of the chapter discusses intersection properties, particularly on the Euclidean property and non-Euclidean oriented matroids.

Page 5 of 452