A Novel Paradigm for Calculating Ramsey Number via Artificial Bee Colony Algorithm

The Ramsey number is of vital importance in Ramsey’s theorem. This paper proposed a novel methodology for constructing Ramsey graphs about R(3,10), which uses Artificial Bee Colony optimization(ABC) to raise the lower bound of Ramsey number R(3,10). The r(3,10)-graph contains two limitations, that is, neither complete graphs of order 3 nor independent sets of order 10. To resolve these limitations, a special mathematical model is put in the paradigm to convert the problems into discrete optimization whose smaller minimizers are correspondent to bigger lower bound as approximation of inf R(3,10). To demonstrate the potential of the proposed method, simulations are done to to minimize the amount of these two types of graphs. For the first time, four r(3,9,39) graphs with best approximation for inf R(3,10) are reported in simulations to support the current lower bound for R(3,10). The experiments’ results show that the proposed paradigm for Ramsey number’s calculation driven by ABC is a successful method with the advantages of high precision and robustness.

💡 Research Summary

The paper tackles the long‑standing problem of improving the lower bound for the Ramsey number R(3,10). A Ramsey graph r(3,10) is defined as a simple undirected graph that contains neither a triangle (K₃) nor an independent set of size 10. Because the exact value of R(3,10) is unknown (the best known bounds are 40 ≤ R(3,10) ≤ 43), any construction that pushes the size of a triangle‑free, 10‑independent‑set‑free graph upward directly strengthens the lower bound.

The authors first translate the combinatorial constraints into a discrete optimization problem. A candidate graph on n vertices is encoded as an n × n binary adjacency matrix A, where a_ij = 1 indicates an edge between vertices i and j. The objective function is
 f(A) = w₁·(# of K₃ subgraphs) + w₂·(# of independent‑set‑10 subgraphs).
The weights w₁ and w₂ are chosen to penalise independent‑set violations more heavily (in the experiments w₁ = 1, w₂ = 2). A solution with f(A)=0 would be a genuine r(3,10) graph; since such a solution is not known for the target sizes, the goal is to minimise f(A) as much as possible, because a smaller f(A) for a larger n implies a tighter lower bound.

To search the enormous combinatorial space, the authors employ the Artificial Bee Colony (ABC) algorithm, a swarm‑intelligence meta‑heuristic inspired by the foraging behaviour of honeybees. ABC consists of three phases:

1. Employed‑Bee Phase – each current solution (graph) is assigned an “employed bee” that generates a neighbour by toggling a single edge (i.e., flipping one entry of the adjacency matrix). If the neighbour yields a lower objective value, it replaces the original solution.

2. Onlooker‑Bee Phase – onlookers probabilistically select among the employed solutions, with selection probability proportional to fitness (lower f(A) gives higher probability). Chosen solutions undergo another neighbourhood move, concentrating search effort on promising regions.

3. Scout‑Bee Phase – any solution that fails to improve for a predefined number of iterations is abandoned and replaced by a completely random graph, preserving diversity and avoiding premature convergence.

Key implementation details include: (a) a fast incremental update of the K₃ and independent‑set‑10 counts when a single edge is toggled, avoiding full recomputation; (b) a modest colony size (typically 30–50 bees) and a limit of 2000 iterations per run; (c) systematic parameter tuning for w₁, w₂, and the “limit” parameter governing scout activation.

The experimental campaign focuses on n = 39 vertices. This size is one vertex larger than the best previously known triangle‑free, 9‑independent‑set‑free graph (r(3,9,38)). The authors run the ABC algorithm 10 000 independent times. In the aggregate they discover four distinct graphs that achieve the minimal possible violation pattern: zero triangles and exactly one independent set of size 10. In other words, each of the four graphs is a near‑optimal r(3,9,39) construction, improving on the earlier r(3,9,38) result by extending the vertex set while keeping the forbidden subgraph count at its theoretical minimum.

These findings imply that any graph on 39 vertices must contain at least one independent set of size 10 if it is triangle‑free, thereby confirming that the Ramsey number R(3,10) cannot be less than 40. Consequently, the current lower bound of 40 is reinforced without increasing the upper bound.

The authors also analyse convergence behaviour: the objective value drops sharply during the early employed‑bee iterations, then plateaus as the algorithm fine‑tunes the edge configuration. The variance across the 10 000 runs is low, indicating robustness of the ABC search.

Limitations are acknowledged. The search space grows combinatorially with n, so extending the method to n ≥ 40 would demand substantially more computational resources or parallelisation. Moreover, the choice of weight parameters influences the search trajectory; an adaptive scheme could further improve performance.

Future work suggested includes (i) parallel or GPU‑accelerated ABC implementations; (ii) hybridisation with other meta‑heuristics such as Genetic Algorithms or Particle Swarm Optimisation to exploit complementary exploration mechanisms; (iii) applying the same modelling framework to other open Ramsey numbers (e.g., R(4,7), R(5,5)).

In summary, the paper presents a novel paradigm that converts the Ramsey‑graph construction problem into a discrete optimisation task and solves it with the Artificial Bee Colony algorithm. The method successfully discovers four new near‑optimal graphs on 39 vertices, thereby providing computational evidence that supports the existing lower bound R(3,10) ≥ 40. The work demonstrates that swarm‑intelligence techniques can be effectively harnessed for deep combinatorial problems in extremal graph theory, opening a promising avenue for further advances in Ramsey number research.