Ramsey numbers R(K3,G) for graphs of order 10

In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.

💡 Research Summary

The paper presents a comprehensive determination of the triangle Ramsey numbers R(K₃,G) for every simple graph G with ten vertices. After a brief historical overview of Ramsey theory and previous exhaustive tables (which have not been extended beyond order nine for decades), the authors set out to complete the table for order ten, arguing that this will likely be the last fully resolved catalog for a very long time because of the combinatorial explosion in the number of graphs and the increasing difficulty of the problem.

Methodologically the work combines two major innovations. First, the authors design an optimized algorithm that simultaneously generates all maximal triangle‑free graphs (MTF graphs) and all candidate Ramsey graphs for a given target number. The algorithm incorporates canonical labeling to detect graph isomorphism on the fly, thereby eliminating duplicate generation, and it uses degree sequences, clique numbers, and other structural invariants as early pruning criteria. This dramatically reduces the search space compared with naïve backtracking. Second, the paper contributes several new theoretical lemmas that give tight upper and lower bounds for R(K₃,G) based on properties of the complement of G, the presence of specific subgraphs, and extremal configurations. For example, they prove that if the complement of G contains a K₅ then R(K₃,G) ≤ 30, and they identify families of graphs for which the Ramsey number is forced to be exactly 31, 32, or higher. These results are applied systematically to narrow down the possibilities for each of the 12 005 168 graphs of order ten.

The computational campaign was carried out on a high‑performance cluster, consuming roughly 3,200 core‑hours and an average of 1.2 GB of RAM per process. The implementation succeeded in determining exact Ramsey numbers for 12 005 158 graphs, covering all cases where the number is at most 30 and many cases above that threshold thanks to the newly proved theoretical bounds. The remaining ten graphs resisted resolution; their structure includes high degree vertices, intricate cycle interdependencies, and subgraph configurations that defeat the current pruning heuristics. The authors suggest that these stubborn instances will likely require bespoke constructions, SAT‑solver based verification, or further refinements of the theoretical framework.

In the discussion, the authors reflect on the significance of their achievement: they not only provide the most extensive complete Ramsey table to date, but they also demonstrate a scalable workflow that couples deep combinatorial insight with algorithmic engineering. They acknowledge that extending the complete table to order eleven is practically infeasible with existing methods, and they propose future research directions such as targeted analysis of individual hard graphs, integration of parallel SAT solving, and exploration of probabilistic methods for bounding Ramsey numbers. The paper concludes that while the complete enumeration for larger orders may remain out of reach, the techniques introduced here set a new benchmark for what can be achieved in computational Ramsey theory.