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Abstract A transversal in an latin square is a collection of entries not repeating any row, column, or symbol. Kwan showed that almost every latin square has transversals as . Using a loose variant of the circle method we sharpen this to . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups. 

Abstract A family of vectors in [ k ] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [ k ] n invariant under a transitive group of symmetries is o ( k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.