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6914 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 39 of 346

  • New upper bounds for the Erdős-Gyárfás problem on generalized Ramsey numbers

  • Part of
  • Alex Cameron, Emily Heath
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 2 / March 2023
    Published online by Cambridge University Press:
    24 November 2022, pp. 349-362
    • Article
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  • A $(p,q)$-colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$-clique. The problem of determining the minimum number of colours, $f(n,p,q)$, needed to give a $(p,q)$-colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$, each of which was proved by giving a deterministic construction which combined a $(p,p-1)$-colouring using few colours with an algebraic colouring.

    In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$-cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$-colourings, which would otherwise make this problem intractable for large values of $p$. In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$.

  • 6 - Statistical Mechanics

  • from II - Mathematical Tools of Complexity Science
  • Henrik Jeldtoft Jensen, Imperial College London
  • Book:
    Complexity Science
    Published online:
    13 December 2022
    Print publication:
    17 November 2022, pp 110-162

  • Summary

    Properties and behaviours at the systemic aggregate level are derived as statistical averages from probability distributions describing the likelihoods of the various states available to the components.

  • 9 - Information Theory and Entropy

  • from II - Mathematical Tools of Complexity Science
  • Henrik Jeldtoft Jensen, Imperial College London
  • Book:
    Complexity Science
    Published online:
    13 December 2022
    Print publication:
    17 November 2022, pp 230-278

  • Summary

    Assume we are able to obtain the joint probability for a set of time series representing a complex system. Based on the joint probabilities, information theory can help to analyse the nature of the interdependence in the system. It is particularly important to be able to distinguish between different types of emergent behaviour, such as synergy or redundancy.

Page 39 of 346