 One of Ajtai's results states that the length of proofs in propositional logic of the pigeonhole principle for "n" items grows faster than any polynomial in "n".He also proved that the statement "any two countable structures that are second-order equivalent are also isomorphic" is both consistent with and independent of ZFC.Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem.