Prabhakar Raghavan

Ansh Nagda, Prabhakar Raghavan, Abhradeep Thakurta

We present improved lower bounds for five 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 $148$, and $\mathbf{R}(4, 15)$ from $158$ to $159$. 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.

Alex Bie, Travis Dick, Alex Kulesza et al.

Modern AI systems have been successfully deployed to win medals at international math competitions, assist with research workflows, and prove novel technical lemmas. However, despite their progress at advanced levels of mathematics, they remain stubbornly bad at basic arithmetic, consistently failing on the simple task of adding two numbers. We present a systematic investigation of this phenomenon. We demonstrate empirically that all frontier models suffer significantly degraded accuracy for integer addition as the number of digits increases. Furthermore, we show that most errors made by these models are highly interpretable and can be attributed to either operand misalignment or a failure to correctly carry; these two error classes explain 87.9%, 62.9%, and 92.4% of Claude Opus 4.1, GPT-5, and Gemini 2.5 Pro errors, respectively. Finally, we show that misalignment errors are frequently related to tokenization, and that carrying errors appear largely as independent random failures.

Ansh Nagda, Prabhakar Raghavan, Abhradeep Thakurta

We explore whether techniques from AI can help discover new combinatorial structures that improve on known limits on efficient algorithms. Specifically, we use AlphaEvolve (an LLM coding agent) to study two settings: a) Average-case hardness for MAX-CUT and MAX-Independent Set: We improve a recent result of Kunisky and Yu to obtain near-optimal upper and (conditional) lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random 3- and 4-regular graphs. Our improved lower bounds are obtained by constructing nearly extremal Ramanujan graphs on as many as $163$ nodes, using AlphaEvolve. Additionally, via analytical arguments we strengthen the upper bounds to settle the computational hardness of these questions up to an error in the third decimal place. b) Worst-case Hardness of Approximation for MAX-k-CUT: We obtain new inapproximability results, proving that it is NP-hard to approximate MAX-4-CUT and MAX-3-CUT within factors of $0.987$ and $0.9649$ respectively, using AlphaEvolve to discover new gadget reductions. Our MAX-4-CUT result improves upon the SOTA of $0.9883$, and our MAX-3-CUT result improves on the current best gadget-based inapproximability result of $0.9853$, but falls short of improving the SOTA of $16/17$ that relies on a custom PCP, rather than a gadget reduction from "standard" Håstad-style PCPs. A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (often requiring exponential time). In both settings above, our results were enabled by using AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000\times$). We conclude with a discussion of norms by which to assess the assistance from AI in developing proofs.

Guan Zhe Hong, Bhavya Vasudeva, Vatsal Sharan et al.

When large language models (LLMs) use in-context learning (ICL) to solve a new task, they must infer latent concepts from demonstration examples. This raises the question of whether and how transformers represent latent structures as part of their computation. Our work experiments with several controlled tasks, studying this question using mechanistic interpretability. First, we show that in transitive reasoning tasks with a latent, discrete concept, the model successfully identifies the latent concept and does step-by-step concept composition. This builds upon prior work that analyzes single-step reasoning. Then, we consider tasks parameterized by a latent numerical concept. We discover low-dimensional subspaces in the model's representation space, where the geometry cleanly reflects the underlying parameterization. Overall, we show that small and large models can indeed disentangle and utilize latent concepts that they learn in-context from a handful of abbreviated demonstrations.