Formalising New Mathematics in Isabelle: Diagonal Ramsey

Abstract: The formalisation of mathematics is starting to become routine, but the value of this technology to the work of mathematicians remains to be shown. There are few examples of using proof assistants to verify brand-new work. This paper reports the formalisation of a major new result (arXiv:2303.09521) about Ramsey numbers that was announced in 2023. One unexpected finding was the heavy need for computer algebra techniques.

• The paper presents a new formalisation that refines the upper bound for Ramsey numbers from 4^k to (4-ε)^k using Isabelle.
• It integrates computer algebra with interactive theorem proving by employing a book algorithm and high-precision real arithmetic in 12,500 lines of code.
• This work demonstrates the potential of proof assistants to verify complex combinatorial proofs and encourages future collaborative research.

Formalising New Mathematics in Isabelle: Diagonal Ramsey

The paper under review presents the formalisation of a recent mathematical breakthrough in the field of Ramsey numbers using Isabelle, a well-known proof assistant. The focus is on a new result in Ramsey theory that achieved a reduction in the upper bound for these numbers, shifting from the classical $(4^k)$ to a more refined $(4-\epsilon)^k$. This formalisation effort represents a significant benchmark in proof assistant applications in verifying new mathematical work before journal publication.

Key Highlights and Numerical Claims

A central theme of this research is the integration of computer algebra with interactive theorem proving to handle the intricate calculations and reasoning required in this domain. The paper elucidates the process and complexity of formalising a lengthy mathematical argument, comprising some 12,500 lines of Isabelle code, dedicated almost entirely to the new result in Ramsey theory.

The main results include:

• Improved bounds for Ramsey numbers when $\ell \leq k/9$ ("far from the diagonal") and $\ell \leq k/4$ ("closer to the diagonal"), transitioning towards the diagonal.
• The introduction of a book algorithm that strategically incorporates degree regularisation and vertex selection to opt for red step, big blue step, or density-boost step, thereby refining the prediction model for Ramsey numbers.
• The utilisation of high-precision real arithmetic, asymptotic reasoning, and sharp bounds essential for verifying claimed results.

The paper also discusses a significant aspect of the formal verification process, which is the requirement for proofs about expectations and edge densities using advanced probabilistic methods combined with graph theory.

Implications and Speculation on Future Developments

The formalisation within Isabelle not only serves to validate the new theoretical advances but also showcases the potential and intricacies of mechanised mathematics. The ability to test hypotheses and choose parameters dynamically within a framework enhances understanding and potentially sharpens the original results.

Future applications could see broader collaboration between mathematicians and computer scientists, where proof assistants like Isabelle serve as genuine 'assistants' rather than mere validators. In practical terms, the ability to maintain a living document that allows for experimentation indicates that proof assistants might soon assist in refining existing proofs or even discovering new ones.

This work also suggests that formal verification can play a broader role beyond theoretical improvements in Ramsey numbers, potentially influencing areas where combinatorial reasoning and probabilistic methods intersect.

Conclusion

This formalisation effort is critical not only for providing unquestionable verification of the new Ramsey numbers result but also for advancing the state of the art in proof assistant technologies. While the field has made significant progress, the journey towards making proof assistants indispensable tools for all mathematicians entails further research and development. This paper stands as a testament to the power of formal methods in mathematics and their role in verifying cutting-edge research.