A New Lower Bound for the Ramsey Number R(4, 8)

The lower bound for the classical Ramsey number R(4, 8) is improved from 56 to 58. The author has found a new edge coloring of K_{57} that has no complete graphs of order 4 in the first color, and no complete graphs of order 8 in the second color. The coloring was found using a SAT solver which is based on MiniSat and customized for solving Ramsey problems.

💡 Research Summary

The paper addresses the long‑standing problem of improving the lower bound for the classical two‑color Ramsey number R(4,8). The author demonstrates that the bound can be raised from 56 to 58 by constructing a two‑color edge‑coloring of the complete graph K₅₇ that contains no monochromatic K₄ in the first color and no monochromatic K₈ in the second color.

The methodology begins with the standard encoding of the Ramsey problem into a Boolean satisfiability (SAT) instance. For a given n, each unordered pair of vertices (i, j) is represented by a Boolean variable eᵢⱼ that is true when the edge is colored with the first color and false otherwise. The requirement “no Kₛ in color 1” and “no Kₜ in color 2” translates into a conjunctive normal form (CNF) consisting of clauses that forbid all‑true assignments on the edges of any s‑set and all‑false assignments on the edges of any t‑set. Directly encoding these constraints leads to O(n²) variables and O(n·s + n·t) clauses, which quickly becomes intractable for the sizes of interest.

To dramatically reduce the search space, the author introduces a set of auxiliary “Z‑variables” zₖ (1 ≤ k ≤ n‑1) that capture the color of edges solely based on the distance k = j‑i (mod n). The constraint eᵢⱼ ↔ z_{j‑i} is expressed by two clauses per edge, effectively collapsing the n(n‑1)/2 edge variables to only n‑1 Z‑variables. This “Z‑constraint” forces the coloring to be highly symmetric (circulant), which both simplifies the SAT instance and preserves many useful structural properties of Ramsey graphs.

While the full Z‑constraint works for several known Ramsey graphs (e.g., a (4,7)‑coloring of K₄₆), it is too restrictive for larger parameters; for instance, no (4,7)‑graph exists with n > 46 under the strict Z‑constraint. To overcome this limitation, the author designs a dynamic relaxation scheme. During the SAT search, each Z‑clause that causes a conflict receives a penalty score. After a predetermined number of failures, the algorithm removes a subset of high‑penalty Z‑clauses, restarts the search, and repeats the process. This approach allows partial retention of the symmetry while giving the solver enough freedom to find a solution. Additional relaxation strategies—such as omitting specific Z‑clauses (“imperfect Z”) or partitioning the Z‑variables according to row/column patterns (“partitioned Z”)—are also explored.

The experimental pipeline proceeds as follows. First, a known (4,7)‑coloring of K₄₈ (Figure 2) is fixed; all edges among the first 48 vertices retain their colors. The remaining edges (those involving vertices 48…56) are left undecided. The SAT instance for these unsettled edges consists of 468 Boolean variables, 56 Z‑variables, roughly 3.48 million Ramsey clauses, and 936 Z‑clauses. The solver, a customized MiniSat variant, runs for about 1.87 × 10⁶ seconds (≈21 days) on a standard desktop (Intel Core i7, 8 GB RAM). Two rounds of Z‑clause relaxation are performed, each discarding about 50 % of the remaining Z‑clauses, ultimately retaining only about 25 % of the original Z‑constraints. The final model yields a complete two‑coloring of K₅₇ that satisfies the Ramsey conditions, which is displayed in Figure 3 and listed as an adjacency list in the appendix (Figure 4).

The existence of this coloring proves that R(4,8) ≥ 58, improving the previous best lower bound of 56. The paper also provides a table (Table 1) summarizing the largest n for which Z‑Ramsey graphs R_Z(s,t,n) have been found for various (s,t) pairs, illustrating the effectiveness of the Z‑constraint approach across a range of parameters.

In the discussion, the author notes that the method scales well: the specialized program computes the necessary clauses on the fly, avoiding the memory overhead of storing the full CNF. This enables handling instances with n > 100 within reasonable CPU time. The paper concludes by pointing readers to a dedicated website where the source code, detailed data, and additional Ramsey graphs (including the new (4,8)‑graph) are available. Future work is suggested in extending the technique to other unresolved Ramsey numbers, refining the relaxation heuristics, and exploring alternative symmetry constraints that may yield even larger lower bounds.

Overall, the work showcases how modern SAT solving, combined with clever symmetry exploitation and adaptive constraint relaxation, can push the frontier of combinatorial mathematics, delivering a concrete improvement in the long‑standing problem of determining Ramsey numbers.