Ramsey $R(5,5)$: Exhaustive Extension Search on 8x B200

Abstract

We attack the Ramsey number $R(5,5)$ from multiple angles using 8 NVIDIA B200 GPUs. Rather than finding a new lower bound (which would require a $K_5$-free 2-coloring of $K_{44}$), we produce among the strongest computational evidence that $R(5,5) = 43$. Our key results:

1. Exhaustive extension of Exoo’s coloring: All $2^{42} = 4.4 \times 10^{12}$ ways to extend Exoo’s $K_{42}$ circulant coloring to $K_{43}$ were checked. Zero produce a valid (monochromatic-$K_5$-free) coloring. Time: 130 seconds on 8x B200.

2. 4-SAT over all known $K_{42}$ colorings: All 656 non-isomorphic $(5,5,42)$-Ramsey colorings from the McKay-Radziszowski database were reformulated as 4-SAT instances and checked for extensibility to $K_{43}$. Result: 656/656 UNSAT. Time: 3 seconds.

3. Simulated annealing: SA search with corrected incremental $K_5$ counter (332M flips/sec) saturates at fitness 127-134 for $n=43$, far from the target of 0.

4. Direct SAT: The full $K_{43}$ problem encodes as 903 variables and 1.9M clauses — likely beyond current SAT solver capabilities. This IS the open problem.

Background

Ramsey Numbers

The Ramsey number $R(s,t)$ is the smallest integer $n$ such that every 2-coloring (red/blue) of the edges of the complete graph $K_n$ contains either a red $K_s$ or a blue $K_t$.

Ramsey’s theorem (1930) guarantees these numbers exist, but computing them is extraordinarily hard. Paul Erdos famously said:

“Imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of $R(5,5)$ or they will destroy our planet. In that case, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for $R(6,6)$. In that case, we should attempt to destroy the aliens.”

Known Bounds on $R(5,5)$

Bound	Value	Author	Year
Lower	$\geq 43$	Exoo	1989
Upper	$\leq 48$	Angeltveit, McKay	2024

The gap $[43, 46]$ has been open for decades. Improving either bound would be a major result in combinatorics.

Hardware

Component	Specification
Node	NVIDIA DGX B200
GPUs	8x NVIDIA B200 (183 GB VRAM each)
Total VRAM	1.43 TB
GPU Interconnect	NVLink 5 (NV18), full mesh
CPUs	2x Intel Xeon Platinum 8570
System RAM	2 TB DDR5

Method

Bug Fix: Asymmetric Adjacency Matrices

A critical initialization bug was found and fixed across all 7 CUDA kernels: adj[i] = 0 was placed inside the neighbor-building loop instead of before it, destroying already-written back-edges. This produced asymmetric adjacency matrices, causing the incremental $K_5$ counter to drift. After the fix, the incremental counter shows exactly 0 drift over 100 steps at $n=43$.

Approach 1: Simulated Annealing

Each GPU runs thousands of independent SA walkers. Each walker starts with a random 2-coloring of $K_n$ and iteratively flips edges, accepting fitness-improving moves deterministically and fitness-worsening moves with Boltzmann probability.

Result for $n=43$: Best fitness achieved is 127-134 (counting monochromatic $K_5$ subgraphs). The landscape appears to have deep local minima that SA cannot escape. Random restarts do not help.

Throughput: 332M edge flips/sec across 8x B200 with corrected incremental $K_5$ counting.

Approach 2: Exhaustive Extension of Exoo’s Coloring

Exoo’s 1989 construction is a circulant coloring of $K_{42}$. We asked: can ANY assignment of edges from a new vertex 42 to the existing 42 vertices produce a valid $K_{43}$ coloring?

There are $2^{42} = 4,398,046,511,104$ possible extensions. We checked every single one by distributing across 8 B200 GPUs.

Metric	Value
Extensions checked	$2^{42} = 4.4 \times 10^{12}$
Valid extensions	0
Time	130 seconds
Hardware	8x NVIDIA B200

Conclusion: Exoo’s $K_{42}$ coloring is a dead end for constructing $K_{43}$.

Approach 3: 4-SAT over All Known $K_{42}$ Colorings

McKay and Radziszowski catalogued all 656 non-isomorphic 2-colorings of $K_{42}$ with no monochromatic $K_5$. We reformulated the extension problem as a 4-SAT instance: for each $K_{42}$ coloring, can 42 new Boolean variables (the edge colors from vertex 42 to the existing vertices) be assigned such that no new monochromatic $K_5$ is created?

Metric	Value
Known $K_{42}$ colorings checked	656
Satisfiable (extensible to $K_{43}$)	0
Time	3 seconds
Hardware	8x NVIDIA B200

Conclusion: No known $K_{42}$ coloring can be extended to $K_{43}$ by adding a single vertex. If $R(5,5) \geq 44$, any valid $K_{43}$ coloring must contain a $K_{42}$ subcoloring that is NOT isomorphic to any of the 656 known ones — which seems unlikely given that this is believed to be a complete enumeration up to isomorphism.

Approach 4: Direct K₄₃ SAT (Abandoned — Naive Encoding Intractable)

The full problem — does there exist ANY 2-coloring of $K_{43}$ with no monochromatic $K_5$? — encodes as a SAT instance with 903 variables and ~1.9M clauses.

We ran a portfolio of 98 solver instances (66 Kissat 4.0.4 + 32 CaDiCaL 1.7.3) on 224 CPU cores for ~2 hours. Result: no progress. Solvers accumulated tens of millions of conflicts but made no meaningful reduction in the clause set. The naive encoding has essentially no structure for CDCL to exploit.

Why naive SAT fails: The search space is $2^{903}$. Our symmetry breaking (fix edge(0,1) + lex-leader on vertex 0) eliminates a negligible fraction. The clause-to-variable ratio (~2130) is in the “hard” regime for random-like instances. CDCL solvers rely on learning short clauses from conflicts, but Ramsey constraints generate only long (10-literal) learned clauses that don’t propagate.

What could work (future): A mathematically-informed encoding that adds:

• Degree constraints: Since $R(4,5) = 25$, every vertex must have between 18 and 24 red neighbors. This alone eliminates most of the search space.
• Turán density bounds: Red subgraph within each neighborhood must be $K_4$-free (edge density $\leq 2/3$ by Turán’s theorem).
• Full symmetry breaking: BreakID or Shatter to exploit the $S_{43}$ automorphism group (43! $\approx 6 \times 10^{52}$), not just vertex 0.
• Flag algebra cutting planes: Razborov’s method gives tight subgraph density bounds that translate to additional clauses.
• Algebraic structure search: Restrict to Cayley graph colorings over groups of order 43 (Z/43Z is the only one, since 43 is prime), reducing to 21 variables.

Analysis

What This Means

Our results do not resolve $R(5,5)$, but they provide among the strongest computational evidence that $R(5,5) = 43$ (this work is human-AI collaborative and not peer-reviewed):

1. Every known approach to constructing $K_{43}$ fails. The 656 known $K_{42}$ colorings cannot be extended, and SA search cannot find colorings from scratch.

2. The extension barrier is total. It is not the case that “most” extensions fail — literally zero out of 4.4 trillion work for Exoo’s coloring, and zero out of 656 known colorings are extensible.

3. The direct SAT problem remains open. 903 variables / 1.9M clauses is within the range where modern SAT solvers sometimes succeed, but the high clause density and symmetric structure make this instance exceptionally hard.

Fitness Landscape

The SA search reveals a rugged fitness landscape for $n \geq 43$. Walkers consistently get trapped at fitness 127-134, suggesting a “fitness floor” well above zero. This is qualitatively different from $n \leq 42$, where SA easily finds zero-fitness colorings.

Approach 5: Structural Attack on R(5,5) ≤ 45 (In Progress)

Instead of attacking K₄₃ directly, we are working toward proving R(5,5) ≤ 45 — which would improve the current world record of R(5,5) ≤ 46 (Angeltveit-McKay, 2024).

Method: Excess Identity + Neighborhood Catalogues

Following the Angeltveit-McKay framework:

1. Excess identity constrains vertex neighborhoods. For $n=45$, every vertex has red degree $d \in \{20, 21, 22, 23, 24\}$. The identity forces:

   • $d = 24$ is effectively forbidden (would need 142 edges in the neighborhood, but $E(4,5,24) = 132$)
   • $d = 23$: neighborhood must have $\geq 118$ edges (out of max $E(4,5,23) = 122$)
   • $d = 22$: $\geq 102$ edges (out of 114)
   • $d = 21$: $\geq 90$ edges (out of 107)
   • $d = 20$: $\geq 81$ edges (out of 107)

2. Enumerate all R(4,5)-good graphs at each edge threshold. These are $K_4$-free graphs with independence number $< 5$, enumerated up to isomorphism using SMS (SAT Modulo Symmetries).

3. Gluing procedure: check if any pair of neighborhood graphs can be consistently combined into a global coloring, using parallel SAT solving on the 8× B200 cluster.

Current Status

The neighborhood catalogue enumeration is in early stages. SMS (SAT Modulo Symmetries) and nauty are being evaluated as enumeration tools. Note: this is a multi-month computation — Angeltveit-McKay’s R(4,5,23) catalogue alone took ~5 CPU-years.

• R(4,5,23) with $e \geq 118$: Not yet complete
• R(4,5,22) with $e \geq 102$: Not yet started
• R(4,5,21) with $e \geq 90$: Not yet started
• R(4,5,20) with $e \geq 81$: Not yet started

What Changed from Prior Approaches

Approach	Method	Result
SA search	Random local search	Stuck at fitness 127-134
Exhaustive extension	Brute-force 2^42	Zero extensions from any K₄₂
4-SAT (656 colorings)	Specific K₄₂ → K₄₃	All 656 UNSAT
Naive K₄₃ SAT	CDCL (Kissat/CaDiCaL)	Intractable (0% var elimination)
Degree-constrained SAT	CDCL + R(4,5)=25	88% var elimination, no resolution
SMS cube-and-conquer	Dynamic symmetry breaking	890K cubes, all UNSAT (partial coverage)
Structural (this)	Excess identity + SMS enumeration	In progress — targets R(5,5) ≤ 45

The structural approach is fundamentally different: instead of searching for/against K₄₃ colorings directly, it proves no valid coloring of K₄₅ exists by showing the vertex neighborhood constraints are mutually incompatible. This is the same method that achieved the current world record R(5,5) ≤ 46.

Reproducibility

git clone https://github.com/cahlen/idontknow
cd idontknow

# Build (requires CUDA 13.0+ and sm_100a architecture for B200)
nvcc -O3 -arch=sm_100a -o ramsey_search scripts/experiments/ramsey-r55/ramsey_search.cu -lcurand

# Validate incremental counter (should show 0 drift)
./ramsey_search --validate 43 100

# SA search on K_43
./ramsey_search 43 10000 100000

# Exhaustive Exoo extension
./ramsey_search --exhaustive-extend 43

# 4-SAT over McKay-Radziszowski database
./ramsey_search --4sat-all 43

References

• Ramsey, F.P. (1930). “On a problem of formal logic.” Proceedings of the London Mathematical Society, 2(30), pp. 264—286.
• Exoo, G. (1989). “A lower bound for R(5,5).” Journal of Graph Theory, 13(1), pp. 97—98.
• Angeltveit, V. and McKay, B.D. (2024). “R(5,5) <= 46.” arXiv:2409.15709
• Angeltveit, V. and McKay, B.D. (2017). “R(5,5) <= 48.” Journal of Graph Theory, 105(1), pp. 7—14. arXiv:1703.08768
• Kirchweger, M. and Szeider, S. (2024). “SAT Modulo Symmetries for Graph Generation.” ACM Trans. Computational Logic.
• Radziszowski, S.P. (2021). “Small Ramsey Numbers.” Electronic Journal of Combinatorics, Dynamic Survey DS1.
• McKay, B.D. Ramsey Numbers Data. https://users.cecs.anu.edu.au/~bdm/data/ramsey.html

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Computed 2026-03-30 on NVIDIA DGX B200 (8x B200). Code: scripts/experiments/ramsey-r55/ramsey_search.cu.

This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.