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A lower bound for set-coloring Ramsey numbers.

Lucas Aragão1, Maurício Collares2, João Pedro Marciano1

  • 1Instituto de Matemática Pura e Aplicada Rio de Janeiro Brazil.

Random Structures & Algorithms
|March 22, 2024
PubMed
Summary
This summary is machine-generated.

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This study introduces a new random coloring to determine the set-coloring Ramsey number for various parameters. The findings provide tighter bounds, advancing understanding in Ramsey theory.

Area of Science:

  • Graph Theory
  • Combinatorics
  • Ramsey Theory

Background:

  • The set-coloring Ramsey number involves assigning sets of colors to graph edges.
  • Previous research established bounds for specific cases () and recent work for general .

Purpose of the Study:

  • To determine the set-coloring Ramsey number for a broad range of parameters.
  • To improve existing upper and lower bounds, particularly in the range.

Main Methods:

  • Introduction of a novel random coloring technique.
  • Application of this coloring to analyze monochromatic cliques in complete graphs.

Main Results:

  • The new method determines up to polylogarithmic factors for many parameter combinations.
Keywords:
Ramsey theoryprobabilistic methodrandom graphs

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  • Significant improvement over existing bounds in the range.

    • Conclusions:

    • The developed random coloring is effective for analyzing set-coloring Ramsey numbers.
    • This research advances the understanding of Ramsey properties in edge-colored graphs.