Some results on small ordered and cyclic Ramsey numbers

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.

• The paper presents SAT and reinforcement learning techniques to compute exact small ordered and cyclic Ramsey numbers for several graph classes.
• It establishes new bounds and explicit formulas for monotone paths, cycles, stars, nested matchings, and alternating paths.
• The work unifies ordered, cyclic, and permutational Ramsey frameworks, offering computational benchmarks, conjectures, and insights for future research.

Summary of "Some results on small ordered and cyclic Ramsey numbers" (2604.16188)

Introduction and Motivation

This paper advances the study of small ordered and cyclic Ramsey numbers for specific classes of graphs, employing SAT solvers and reinforcement learning to find exact values and bounds. Ordered Ramsey numbers require the monochromatic subgraph to be embedded respecting a total ordering; cyclic Ramsey numbers relax this to cyclic ordering, yielding new variants in combinatorial Ramsey theory. The authors focus on two-color Ramsey numbers, primarily for monotone paths, cycles, alternating paths, stars, complete graphs, and nested matchings. Their methodology leverages the Kissat SAT solver and RLGT reinforcement learning framework, enabling computational results for previously unresolved cases.

Definitions and Conceptual Framework

The paper introduces distinct types of Ramsey numbers:

• Ordered Ramsey number $R_\text{ord}(H_1,\ldots,H_k)$: Smallest $n$ such that every $k$-edge coloring of $K_n$ contains a monochromatic copy of $H_j$ in color $j$, with the embedding respecting the vertex order.
• Cyclic Ramsey number $R_\text{cyc}(H_1,\ldots,H_k)$: Smallest $n$ under $k$-edge coloring of $K_n$ where some monochromatic $n$0 appears in color $n$1 under cyclic order (up to rotation).
• Permutational Ramsey numbers: Unification wherein the required embedding respects a subgroup of the symmetric group, enabling generalization to ordered, standard, cyclic, reflective, dihedral, and alternating Ramsey numbers.

The authors extend classical results for monotone paths, ordered cycles, and alternating paths, illustrating how these number variants nest within a lattice of permutation symmetries.

SAT Approach for Ramsey Computation

A reduction to Boolean satisfiability enables computational discovery. For a proposed Ramsey number $n$2, clauses encode the non-existence of forbidden monochromatic ordered (or cyclic) subgraphs. Variables represent edge colors and clauses enumerate all admissible embeddings as per the ordering constraints. SAT resolution establishes upper/lower bounds, bifurcated into ordered and cyclic variants via explicitly constructed clause sets.

Reinforcement Learning Methodology

Reinforcement learning, via RLGT, is used for heuristic lower bound discovery. Agents iteratively construct edge colorings of $n$3 aiming to maximize a negative count of forbidden embeddings. When agent reward reaches zero, a coloring avoiding all forbidden monochromatic subgraphs is found, yielding valid lower bounds. Optimality and upper bound computation are not generally accessible via RL. For Ramsey numbers less than $n$4, RLGT can occasionally find competitive lower bounds but is surpassed by SAT-based methods for exact computation.

Main Results and Numerical Findings

The paper provides extensive tables of computed Ramsey numbers for numerous graph classes and combinations thereof. Several theorems establish exact small Ramsey numbers, and conjectures are proposed for cases where only empirical data is available.

Strong Explicit Results

• Monotone Paths: $n$5, for $n$6-color cases.
• Monotone Cycles: $n$7; this value also holds for cyclic Ramsey numbers.
• Connected graphs and Monotone Paths: $n$8 for connected $n$9.
• Stars: Ordered and cyclic Ramsey numbers coincide with classic values from Burr and Roberts, e.g., $k$0.
• Nested Matchings: $k$1.

Numerical Results

Computations extend ordered and cyclic Ramsey number tables for monotone paths, alternating paths, monotone cycles, complete graphs, stars, and nested matchings. The cyclic numbers for many cases are strictly intermediate between ordered and classical Ramsey numbers, sometimes exhibiting sharp asymptotic contrast. The empirical findings enable conjectures for growth rates and precise numbers in myriad cases.

Contradictory/Nontrivial Observations

• For certain pairs, cyclic Ramsey numbers are significantly smaller than ordered versions ($k$2, while the minimum ordered is $k$3).
• Empirical patterns for alternating paths diverge from monotone paths, with conjectured formulas differing by parity and structure.
• In some cases, the gap between standard, cyclic, and ordered numbers is unbounded as parameters grow.

Theoretical and Practical Implications

The unification of Ramsey-type invariants under permutation symmetry provides a generalized framework, linking combinatorial constructions to group-theoretic constraints. Practically, computational SAT and RL techniques produce new values and bounds inaccessible via classical methods. The cyclic Ramsey number exhibits distinct structural and growth properties, inviting further theoretical analysis.

From a computational perspective, SAT-based methods dominate for upper and lower bound determination, especially for ordered scenarios. RL-based methods are promising for heuristic lower bounds and may scale to more complex Ramsey-type questions, though their practical frontier is presently limited.

Speculation on Future Developments

The paper identifies several open conjectures regarding cyclic Ramsey number asymptotics, behavior for nested matchings, and growth patterns for alternating versus monotone paths. The permutational Ramsey number framework enables new lines of inquiry into symmetry-restricted combinatorial existence. There is potential for extending SAT and RL methods to larger-scale and more intricate Ramsey-type problems, including multi-color variants, hypergraphs, and geometric Ramsey theory. Moreover, the interplay between symmetry and extremal graph theory could produce refined bounds and structural results, particularly as computational power increases and RL techniques evolve.

Conclusion

This work advances the computation and understanding of small ordered and cyclic Ramsey numbers, extending classical and contemporary theories via computational techniques. It introduces permutational Ramsey numbers as a unifying abstraction and demonstrates structural differences between ordered, cyclic, and standard Ramsey numbers. The results provide benchmarks, conjectures, and methodologies for ongoing exploration in Ramsey theory and computational combinatorics.