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arxiv: 2604.17684 · v1 · submitted 2026-04-20 · 🧮 math.CO

The Gamma-Switch Ramsey Number

Christopher Duffy , Benjamin Fok , Gary MacGillivray This is my paper

Pith reviewed 2026-05-10 05:07 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C55
keywords Ramsey numbersedge colouringsgroup actions on colourscyclic groupsmulticolour Ramseyswitch Ramsey
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The pith

The Γ-switch Ramsey number equals the classical Ramsey number plus one for certain cyclic groups and parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines the Γ-switch Ramsey number as a variant of the multicolour Ramsey number in which the colours on edges at each vertex can be locally permuted by elements of a fixed group Γ. It develops basic bounds relating the new parameter to the ordinary Ramsey number R and then computes exact values or tight bounds in small cases. The central results show that for the cyclic group of order three the switch version of R(4,4,4) is exactly one larger than the classical number, and similarly for a mixed 4-tuple with the cyclic group of order four; a further bound places the four-colour case between 43 and the classical R(3,3,3,3) plus one. A sympathetic reader cares because the construction isolates the effect of local symmetry on the global Ramsey threshold, showing that such symmetry raises the threshold by a very small additive constant.

Core claim

We prove R_{C_3}(4,4,4) = R(3,3,3) + 1, R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1 and 43 ≤ R_{C_4}(4,4,4,4) ≤ R(3,3,3,3) + 1.

What carries the argument

The Γ-switch Ramsey number, which requires that no colouring admits a monochromatic clique even after any allowed local permutation of colours at each vertex drawn from the group Γ.

If this is right

  • For the cyclic groups examined, introducing local colour switches raises the Ramsey threshold by exactly one vertex.
  • The upper bound shows that the switch variant cannot exceed the classical multicolour Ramsey number by more than one in the four-colour case.
  • The lower bound of 43 for the four-colour cyclic-switch number gives a concrete starting point for further computation.
  • The results suggest that the additive gap between switch and classical Ramsey numbers is independent of the specific group order in these small instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the pattern holds for larger cyclic groups, then computing the switch numbers may be no harder than computing the classical ones plus a trivial check.
  • The construction could be used to obtain new upper bounds on ordinary Ramsey numbers by embedding a classical colouring into a switch colouring on one extra vertex.
  • The same local-permutation idea might apply to other Ramsey-type parameters such as Folkman numbers or Schur numbers.

Load-bearing premise

The chosen group action on the set of colours must be compatible with the edge colouring so that the monochromatic-clique condition is preserved under the permitted local switches.

What would settle it

An explicit 4-edge-colouring of the complete graph on R(3,3,3,3) vertices that admits no monochromatic K_4 after any allowed C_4-permutation of colours at each vertex would falsify the upper bound.

Figures

Figures reproduced from arXiv: 2604.17684 by Benjamin Fok, Christopher Duffy, Gary MacGillivray.

Figure 1
Figure 1. Figure 1: Configurations for j = 1, 3 in the proof of Theorem 2.2 edge of the form vk,γi vk,γj , then H is a subgraph of P(G). And so by Theorem 2.2, G can be Γ-switched to obtain a copy of H and the result follows. Otherwise, assume H contains an edge of the form vk,γi vk,γj . Without loss of generality, assume the following: H contains the edge v1,γ1 v1,γ2 and cP⋆(G)(v1,γ1 v1,γ2 ) = 1. Let x = v1,γ1 and y = v1,γ2 … view at source ↗
read the original abstract

We define and develop preliminary theoretical results for the $\Gamma$-switch Ramsey number, a variation on the classical $m$-colour Ramsey number for which we allow permuting the colours incident with a vertex using elements of a group $\Gamma \leq S_m$. We find bounds for the $\Gamma$-switch Ramsey number for groups with various properties as a function of the classical parameter. We prove $R_{C_3}(4,4,4) = R(3,3,3) + 1$, $R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1$ and $43 \leq R_{C_4}(4,4,4,4) \leq R(3,3,3,3) + 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Γ-switch Ramsey number R_Γ as a variant of the classical m-color Ramsey number in which, for a group Γ ≤ S_m, each vertex v is permitted to apply an independent permutation γ_v ∈ Γ to the colors of all edges incident to v. It derives general bounds relating R_Γ to the ordinary Ramsey number R and proves three concrete statements: R_{C_3}(4,4,4) = R(3,3,3) + 1, R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1, and 43 ≤ R_{C_4}(4,4,4,4) ≤ R(3,3,3,3) + 1.

Significance. If the Γ-switch operation is shown to produce a well-defined edge-coloring whose monochromatic-clique condition is unambiguously preserved, the exact equalities supply new, parameter-free relations between classical Ramsey numbers and their group-action variants; the lower-bound construction for the 4-color C_4 case is also a concrete, falsifiable prediction.

major comments (2)
  1. [Definition of Γ-switch Ramsey number] The definition of the Γ-switch operation (preliminary section) must explicitly reconcile the two independently chosen permutations γ_u(c) and γ_v(c) that an edge {u,v} receives from its endpoints; without a stated rule (global equality, orientation, or local reinterpretation of monochromaticity), it is unclear whether the resulting object remains a proper edge-coloring, which directly affects the validity of the three claimed equalities.
  2. [Proofs of the exact equalities] The proofs of the two exact equalities (R_{C_3}(4,4,4) = R(3,3,3)+1 and R_{C_4}(3,4,3,4)=R(2,3,2,3)+1) rely on the switched coloring preserving the monochromatic-clique property in both directions; the manuscript should supply the precise compatibility condition used in those arguments.
minor comments (2)
  1. Notation for the specific groups C_3 and C_4 should be defined explicitly (e.g., as cyclic subgroups of S_3 and S_4) rather than left implicit.
  2. The lower-bound construction yielding 43 for R_{C_4}(4,4,4,4) would benefit from a brief description of the underlying coloring or reference to a computational verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need for greater precision in the definition and proofs. We will revise the manuscript to clarify the Γ-switch operation and to supply the missing compatibility details, thereby strengthening the presentation of the results.

read point-by-point responses

  1. Referee: The definition of the Γ-switch operation (preliminary section) must explicitly reconcile the two independently chosen permutations γ_u(c) and γ_v(c) that an edge {u,v} receives from its endpoints; without a stated rule (global equality, orientation, or local reinterpretation of monochromaticity), it is unclear whether the resulting object remains a proper edge-coloring, which directly affects the validity of the three claimed equalities.

    Authors: We agree that the preliminary definition requires an explicit reconciliation rule. In the revised manuscript we will add a precise statement: the color of edge {u,v} is interpreted locally via the composition of the two permutations at its endpoints, with monochromaticity defined relative to the permuted color labels at each vertex. This rule ensures the switched object is a well-defined edge-coloring and is consistent with the group action of Γ. The added paragraph will also confirm that the three claimed relations remain valid under this clarification. revision: yes

  2. Referee: The proofs of the two exact equalities (R_{C_3}(4,4,4) = R(3,3,3)+1 and R_{C_4}(3,4,3,4)=R(2,3,2,3)+1) rely on the switched coloring preserving the monochromatic-clique property in both directions; the manuscript should supply the precise compatibility condition used in those arguments.

    Authors: We acknowledge that the proofs would be clearer with an explicit compatibility condition. In the revision we will insert a short lemma immediately before the two equality proofs that states the precise condition: for Γ = C_3 (respectively C_4) the group action preserves monochromatic K_r if and only if the local permutations map the color set of any monochromatic clique to itself. We will then verify that both the upper-bound (ordinary Ramsey) and lower-bound (explicit construction) directions satisfy this condition, thereby making the bidirectional preservation fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: Gamma-switch Ramsey results derived independently from classical parameters

full rationale

The paper defines the new Γ-switch Ramsey number via group actions permitting local color permutations at vertices, then establishes explicit relations and bounds to classical Ramsey numbers (e.g., exact equalities R_{C_3}(4,4,4) = R(3,3,3) + 1 and R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1, plus the interval for the 4-color case). These are presented as theorems obtained from the definition and group properties, without any equations reducing the claimed values to fitted inputs by construction, without self-referential definitions, and without load-bearing self-citations or imported uniqueness results. The derivation introduces an independent variation on edge-colorings and derives its Ramsey-theoretic consequences directly, remaining self-contained against external classical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the new definition of the Γ-switch Ramsey number and on the standard axioms of Ramsey theory and group actions on sets. No free parameters are introduced in the abstract. The only invented entity is the Γ-switch number itself, which is a definition rather than a postulated physical object.

axioms (1)
  • standard math Standard axioms of finite group actions on finite sets and the definition of classical Ramsey numbers.
    Invoked implicitly when relating the new number to R(k1,...,km).
invented entities (1)
  • Γ-switch Ramsey number no independent evidence
    purpose: To measure the minimal n such that every m-edge-colouring of Kn admits a monochromatic clique after any Γ-permutation of colours at each vertex.
    This is the central new definition; it has no independent existence outside the paper.

pith-pipeline@v0.9.0 · 5433 in / 1363 out tokens · 31536 ms · 2026-05-10T05:07:39.209832+00:00 · methodology

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Reference graph

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