Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems

Bryce Alan Christopherson

Paper project · Ramsey theory

Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems

Bryce Alan Christopherson

2025 public preprint · arXiv:2502.04311

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StatusPublic preprint

AimStudy relationships among Ramseyian objects

MethodGeneralized Ramsey numbers and algebraic tools

ExamplesArithmetic progressions and prime-gap statements

Overview

Monochromatic and heterochromatic structure in one framework.

Ramsey theory is often introduced through unavoidable monochromatic structure, but it also has a heterochromatic side: prescribed multicolored and rainbow patterns can become unavoidable. This paper places both sides inside one threshold question—when must every admissible coloring of a growing host contain one of a specified family of colored local patterns?

A Ramsey base specifies the host graphs and admissible global colorings; a Ramsey symbol specifies the target graphs and their local colorings. Classical graph Ramsey numbers return when the hosts are complete graphs, every edge-coloring is allowed, and the targets are monochromatic. The same framework also reaches arithmetic examples: metric colorings on primes turn equal-colored paths into equal successive prime gaps, yielding Ramsey-theoretic formulations of Green–Tao, bounded-gap, Twin Prime, and Polignac-type statements.

Broaden the targets

Include prescribed multicolored patterns

The local structure may be monochromatic, partly multicolored, or rainbow; the colored pattern itself is part of the Ramsey data.

Control the ambient colorings

Choose the admissible global family

Restricting which colorings are allowed lets the same threshold question encode structure beyond unrestricted complete-graph colorings.

Connect distant settings

Expose a common forcing form

Graph-coloring and prime-gap examples become instances of one question about when prescribed colored local structure becomes unavoidable.

Monochromatic and heterochromatic Ramsey theory

The same threshold question, with broader inputs.

Classical Ramsey numbers sit inside a framework where the host sequence, admissible global colorings, and prescribed local colorings are all part of the problem.

The same threshold question, with broader inputsWhen does every allowed coloring contain a prescribed colored pattern?

ClassicalMonochromatic forcing

Host: Complete graphs K_t
Global colorings: Every edge-coloring
Local targets: solid-color cliques
Threshold: R(z_a:a∈A)

GeneralizedPrescribed colored forcing

Host: A hereditary family 𝔾={G_t}
Global colorings: Admissible ρ∈S_t
Local targets: monochromatic, patterned, or rainbow
Threshold: R_(𝔾,𝒮)(𝕏,𝒞)

below thresholda counterexample coloring survives

first unavoidable levelR

all later levelsevery admissible coloring contains a target

The definition quantifies over every host level t≥n. A failed test at level n therefore gives R>n.

Citation

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The citation below uses the arXiv record.

BibTeX citation

@article{christopherson2025ramseyian,
  title         = {Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems},
  author        = {Christopherson, Bryce Alan},
  year          = {2025},
  eprint        = {2502.04311},
  archivePrefix = {arXiv},
  primaryClass  = {math.CO},
  url           = {https://arxiv.org/abs/2502.04311}
}