Direct LLM Prompting for Extremal Combinatorial Discovery

Determine whether direct prompting of large language models can discover extremal combinatorial structures, rather than relying on program-search or code-mutation frameworks such as AlphaEvolve, for example by producing explicit constructions of graphs witnessing lower bounds for classical graph Ramsey numbers.

Background

The paper applies AlphaEvolve, a code-mutation agent guided by LLMs, to discover search algorithms that construct graphs achieving new and matched lower bounds for several classical graph Ramsey numbers. In this paradigm, the LLM iteratively evolves programs that, in turn, search for extremal combinatorial objects.

The authors contrast this approach with recent AI results in theoretical computer science where new theorems have been discovered via direct prompting of LLMs, without a program-evolution loop. They explicitly note uncertainty about whether such direct prompting can yield extremal combinatorial constructions, highlighting a methodological open question about how best to leverage LLMs for discovering structured witnesses in extremal combinatorics.