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arxiv: 2604.16188 · v1 · submitted 2026-04-17 · 🧮 math.CO · cs.DM

Recognition: unknown

Some results on small ordered and cyclic Ramsey numbers

Dragan Stevanovi\'c, Ivan Damnjanovi\'c, Ivan Sto\v{s}i\'c, Nino Ba\v{s}i\'c

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:55 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords ordered ramsey numberscyclic ramsey numbersmonotone pathsmonotone cyclesramsey theorygraph colorings
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The pith

Computational patterns in small ordered and cyclic Ramsey numbers yield exact values for entire classes of graphs such as monotone paths and cycles.

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

The paper computes small two-color ordered Ramsey numbers for classes including monotone paths, monotone cycles, alternating paths, stars, complete graphs, and nested matchings. It introduces cyclic Ramsey numbers as a relaxation of the ordered version. Structural patterns observed in the results allow exact determinations of all such numbers for several pairs of these classes. Bounds are obtained when one graph is connected and the other is a monotone path or cycle. A unification of the standard, ordered, and cyclic versions is proposed through permutational Ramsey numbers.

Core claim

By computing small instances and identifying recurring structural patterns, all ordered or cyclic Ramsey numbers are determined for several pairs of graph classes, such as monotone paths against monotone cycles. Additional bounds are derived for cases where one argument is a connected graph and the other is a monotone path or cycle. Permutational Ramsey numbers are introduced to place the ordered, cyclic, and classical variants inside a single group-theoretic framework.

What carries the argument

SAT encodings of the conditions for avoiding monochromatic ordered or cyclically ordered subgraphs, followed by manual detection of patterns across the resulting tables to obtain closed forms for infinite families.

If this is right

  • Exact closed-form expressions are now available for the ordered Ramsey numbers between any two monotone paths.
  • The same closed forms hold for the corresponding cyclic Ramsey numbers in those classes.
  • Upper and lower bounds apply to all ordered and cyclic Ramsey numbers mixing an arbitrary connected graph with a monotone path or cycle.
  • The permutational formulation places classical, ordered, and cyclic Ramsey numbers inside one common setting.

Where Pith is reading between the lines

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

  • The same pattern-spotting method from small computations could be tried on other families of extremal numbers to generate conjectured formulas.
  • The group-theoretic unification opens the possibility of algebraic proofs that replace the computational evidence for the observed formulas.
  • Because ordered versions require stricter conditions than classical Ramsey numbers, the exact values here supply new lower bounds on classical numbers for the same graphs.

Load-bearing premise

The structural patterns visible in the computed small instances continue to hold for all larger graphs within the same classes.

What would settle it

An explicit counterexample computation for a pair of larger monotone paths whose ordered Ramsey number differs from the formula suggested by the pattern in smaller cases.

Figures

Figures reproduced from arXiv: 2604.16188 by Dragan Stevanovi\'c, Ivan Damnjanovi\'c, Ivan Sto\v{s}i\'c, Nino Ba\v{s}i\'c.

Figure 1
Figure 1. Figure 1: The alternating path graphs of orders five and six. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The reverse alternating path graphs of orders six and seven. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The nested matching graphs of orders four and eight. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph H from Example 6.5. We conclude the paper with the following example, which demonstrates that each inequality from (5) can be strict. Example 6.5. The src/sat_solving folder in [9] contains a Python script for generating SAT instances, which supports reflective and dihedral Ramsey numbers in addition to the ordered and cyclic ones. By exe￾cuting this script several times together with Kissat, it … view at source ↗
read the original abstract

Let $k \in \mathbb{N}$ and let $H_1, H_2, \ldots, H_k$ be simple graphs such that for each $j \in \{ 1, 2, \ldots, k \}$, the vertex set of $H_j$ is $\{ 0, 1, 2, \ldots, n_j - 1 \}$ for some $n_j \in \mathbb{N}$. The ordered Ramsey number $R_\mathrm{ord}(H_1, H_2, \ldots, H_k)$ is the smallest $n \in \mathbb{N}$ for which every $k$-edge-coloring of the complete graph on the vertex set $\{ 0, 1, 2, \ldots, n - 1 \}$ contains $H_j$ as a monochromatic subgraph of color $j$ for some $j \in \{ 1, 2, \ldots, k \}$, with the vertices appearing in the same order as in $H_j$. Inspired by the work of Poljak, we apply the Kissat SAT solver to determine new small two-color ordered Ramsey numbers of various classes of graphs: monotone paths, monotone cycles, alternating paths, stars, complete graphs and nested matchings. In addition, we introduce the cyclic Ramsey numbers $R_\mathrm{cyc}(H_1, H_2, \ldots, H_k)$ as a natural relaxation of the ordered Ramsey numbers, and once again use Kissat to determine various such numbers for the two-color case. By observing structural patterns in the computational results, we determine all ordered or cyclic Ramsey numbers for several pairs of classes of graphs. Furthermore, we obtain some bounds on ordered and cyclic Ramsey numbers where one argument is a connected graph, while the other is a monotone path or a monotone cycle. We also explore how reinforcement learning can be used through the recently developed Reinforcement Learning for Graph Theory (RLGT) framework to obtain lower bounds on ordered and cyclic Ramsey numbers. Finally, we introduce the permutational Ramsey numbers to show how the different Ramsey-type formulations involving standard, ordered and cyclic Ramsey numbers can be unified within a group-theoretic framework.

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 paper defines the ordered Ramsey number R_ord(H1,...,Hk) as the smallest n such that every k-edge-coloring of the ordered complete graph on {0,...,n-1} contains a monochromatic ordered copy of each Hj in color j. It applies the Kissat SAT solver to compute new small two-color ordered Ramsey numbers for classes including monotone paths, monotone cycles, alternating paths, stars, complete graphs, and nested matchings. The paper introduces cyclic Ramsey numbers R_cyc as a relaxation of the ordered variant and computes small values for the same classes. By identifying structural patterns in the computed tables, it claims closed-form determinations for all ordered or cyclic Ramsey numbers in several pairs of graph classes, provides bounds when one argument is a connected graph and the other a monotone path or cycle, explores reinforcement learning via the RLGT framework for lower bounds, and introduces permutational Ramsey numbers to unify standard, ordered, and cyclic variants in a group-theoretic setting.

Significance. If the SAT computations are correct, the work supplies new exact small values and closed-form expressions for ordered and cyclic Ramsey numbers across multiple graph families, extending classical Ramsey theory with order and cyclic constraints. The pattern-based closed forms and the group-theoretic unification via permutational numbers provide structural insight, while the RL lower-bound experiments illustrate an alternative computational technique. These contributions could support further exact determinations and theoretical analysis in structured Ramsey problems.

major comments (2)
  1. [Computational sections (around the Kissat experiments)] The sections describing the computational approach and SAT encodings provide no explicit CNF variable mapping or clause list for the order-preserving embedding condition (for R_ord) or the rotational invariance condition (for R_cyc). Without this, or any independent verification such as reproduction of known small classical Ramsey numbers or hand-checked instances for n≤10, the reported tables cannot be confirmed to correctly model the definitions.
  2. [Sections deriving closed forms from computational results] The closed-form claims for entire classes (e.g., all R_ord or R_cyc between monotone paths and cycles, or similar pairs) rest on patterns observed in the SAT-derived tables. Because the tables themselves lack cross-checks or encoding validation, these extrapolations are load-bearing but rest on unverified data; a single encoding error would invalidate the pattern-based determinations.
minor comments (2)
  1. The abstract mentions the use of Kissat and RLGT but gives no indication of the scale of the computations or the range of graph sizes considered; adding a brief summary of the largest instances solved would improve clarity.
  2. Notation for the new permutational Ramsey numbers is introduced late; an early diagram or table comparing the three variants (standard, ordered, cyclic) would help readers track the unification argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying areas where additional transparency would strengthen the presentation of our computational results. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Computational sections (around the Kissat experiments)] The sections describing the computational approach and SAT encodings provide no explicit CNF variable mapping or clause list for the order-preserving embedding condition (for R_ord) or the rotational invariance condition (for R_cyc). Without this, or any independent verification such as reproduction of known small classical Ramsey numbers or hand-checked instances for n≤10, the reported tables cannot be confirmed to correctly model the definitions.

    Authors: We agree that the current manuscript lacks sufficient detail on the SAT encoding. In the revised version we will add an explicit description of the variable mapping (one Boolean variable per potential edge in each color together with auxiliary variables for vertex mappings) and the principal clauses that enforce order preservation for R_ord and rotational invariance for R_cyc. We will also include a short verification subsection that reproduces several well-known small classical Ramsey numbers (e.g., R(3,3)=6, R(3,4)=9) using the same encoding framework, and we will supply hand-checkable instances for selected n≤10 cases that match the definitions directly. revision: yes

  2. Referee: [Sections deriving closed forms from computational results] The closed-form claims for entire classes (e.g., all R_ord or R_cyc between monotone paths and cycles, or similar pairs) rest on patterns observed in the SAT-derived tables. Because the tables themselves lack cross-checks or encoding validation, these extrapolations are load-bearing but rest on unverified data; a single encoding error would invalidate the pattern-based determinations.

    Authors: The closed-form statements are indeed pattern-based extrapolations from the computed tables. Once the encoding details and verification results are added (as described in the response to the first comment), the reliability of the underlying data will be established. In the revision we will also state clearly which closed forms are accompanied by combinatorial proofs and which remain conjectural but are supported by exhaustive computation up to a verified bound together with matching theoretical upper and lower bounds derived in the paper. This makes the evidential basis explicit rather than load-bearing on unverified tables. revision: partial

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper computes small ordered and cyclic Ramsey numbers via direct Kissat SAT encodings of the embedding conditions, then observes patterns in those independent computational tables to conjecture closed forms for graph classes. No step reduces a claimed result to its own inputs by construction: the solver outputs are external, the pattern step is observational generalization rather than a fitted prediction or self-definition, and no load-bearing self-citations or imported uniqueness theorems appear. New concepts (cyclic, permutational Ramsey numbers) are introduced by explicit definition without circular reference back to the computed values or patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The work rests on the standard definitions of Ramsey numbers and graph homomorphisms, introduces two new entities (cyclic and permutational Ramsey numbers) without external falsifiable evidence, and relies on the correctness of the SAT encoding and the representativeness of the enumerated cases for the pattern-based claims.

axioms (1)
  • standard math Standard definitions of ordered subgraphs and monochromatic copies in edge colorings of complete graphs
    Invoked throughout the definitions of R_ord and R_cyc.
invented entities (2)
  • Cyclic Ramsey number R_cyc(H1,...,Hk) no independent evidence
    purpose: Natural relaxation of the ordered Ramsey number that ignores linear order but retains cyclic structure
    Defined in the paper as a new concept; no independent evidence supplied beyond the computational tables.
  • Permutational Ramsey numbers no independent evidence
    purpose: Group-theoretic unification of standard, ordered and cyclic Ramsey numbers
    Introduced to show how the three formulations sit inside a single framework; no external verification provided.

pith-pipeline@v0.9.0 · 5730 in / 1477 out tokens · 34590 ms · 2026-05-10T07:55:14.154992+00:00 · methodology

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