Reinforced Generation of Combinatorial Structures: Ramsey Numbers

We present improved lower bounds for nine classical Ramsey numbers: R(3, 13) is increased from 60 to 61, R(3, 18) from 99 to 100, R(4, 13) from 138 to 139, R(4, 14) from 147 to 148, R(4, 15) from 158 to 159, R(4, 16) from 170 to 174, R(4, 18) from 205 to 209, R(4, 19) from 213 to 219, and 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.