The vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime p indexed by sum(ki-1).
Reinforced Generation of Combinatorial Structures: Ramsey Numbers
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We present improved lower bounds for nine classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, $\mathbf{R}(4, 15)$ from $158$ to $159$, $\mathbf{R}(4, 16)$ from $170$ to $174$, $\mathbf{R}(4, 18)$ from $205$ to $209$, $\mathbf{R}(4, 19)$ from $213$ to $219$, and $\mathbf{R}(4, 20)$ from $234$ to $237$. These results were achieved using AlphaEvolve, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.
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Litvak's conjecture on minimizing moments of Gaussian minima is disproved by a cosine-based correlation matrix for small n and p, with a new conjecture proposed that this matrix is the general minimizer, supported conditionally via a volumetric extension of Fejes Tóth's zone conjecture.
k-server-bench formulates potential-function discovery for the k-server conjecture as a code-based inequality-satisfaction task; current agents fully solve the resolved k=3 case and reduce violations on the open k=4 case.
pAI/MSc is a customizable multi-agent system that reduces human steering by orders of magnitude when turning a hypothesis into a literature-grounded, mathematically established, experimentally supported manuscript draft in ML theory.
citing papers explorer
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Prime Certificates for Exact Vertex-Coprime Ramsey Numbers
The vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime p indexed by sum(ki-1).
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A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture
Litvak's conjecture on minimizing moments of Gaussian minima is disproved by a cosine-based correlation matrix for small n and p, with a new conjecture proposed that this matrix is the general minimizer, supported conditionally via a volumetric extension of Fejes Tóth's zone conjecture.
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$k$-server-bench: Automating Potential Discovery for the $k$-Server Conjecture
k-server-bench formulates potential-function discovery for the k-server conjecture as a code-based inequality-satisfaction task; current agents fully solve the resolved k=3 case and reduce violations on the open k=4 case.
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pAI/MSc: ML Theory Research with Humans on the Loop
pAI/MSc is a customizable multi-agent system that reduces human steering by orders of magnitude when turning a hypothesis into a literature-grounded, mathematically established, experimentally supported manuscript draft in ML theory.
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