A subset of points in a transversal design is a thwart if each block in the design has one of a small number of intersection sizes with the subset. While the details are too complicated to state here, the authors study thwarts (subconfigurations) of transversal deisgns via the use of a version of Wilson's theorem. As a consequence they are able to obtain improved bounds for \(N(n)\), the maximum number of mutually orthogonal latin squares of order \(n\), for 25 values of \(n\leq 8096\). The connection between transversal designs and sets of orthogonal latin squares comes from the well-known result: \(N(n)\geq k\) if and only if there exists a transversal design \(\text{TD}(k+ 2; n)\).