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The Pigeonhole Principle (PHP) has been heavily studied in automatedreasoning, both theoretically and in practice. Most solvers have exponentialruntime and proof length, while some specialized techniques achieve polynomialruntime and proof length. Several decades ago, Cook manually constructed$O(n^4)$ extended resolution proofs, where $n$ denotes the number ofpigeons.Existing automated techniques only surpass Cook's proofs in similarproof systems for large $n$. We construct the shortest known proofs of PHP inthe standard proof format of modern SAT solving, DRAT. Using auxiliaryvariables and by recursively decomposing the original program into smallersizes, we manually obtain proofs having length $O(n^3)$ and leading coefficient$5/2$.

Towards the shortest DRAT proof of the Pigeonhole Principle