An AI model just solved a mathematical mystery about complex graph patterns that had been open since 1982.
April 24, 2026
Original Paper
Doubly Saturated Ramsey Graphs: A Case Study in Computer-Assisted Mathematical Discovery
arXiv · 2604.21187
The Takeaway
Researchers combined a logic solver with code generated by a language model to find infinite families of specific graphs. This 40-year-old problem in combinatorics had resisted solutions from the best human mathematicians for decades. The AI didn't just guess the answer, it provided the precise code and logic needed for a formal proof. This marks a shift from using AI for casual conversation to using it for high-precision scientific discovery. It demonstrates that the next generation of mathematical breakthroughs will likely involve a hybrid of human intuition and AI-driven logic. Our most stubborn academic puzzles are finally within reach.
From the abstract
Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to genera