which attention centered on the restricted problem. However, with some additional smoothness assumptions on G and M, it might yet be possible to prove the existence of a real analytic structure on the group action. The Hilbert–Smith conjecture, still unsolved, encapsulates the remaining difficulties of this case. Quantum mechanics The Born rule states that an observable property of a quantum system is defined by a Hermitian operator on a separable Hilbert space, that the only observable values of the property are the eigenvalues of the operator, and that the probability of the system being observed in a particular eigenvalue is the square of the absolute value of the complex number obtained by projecting the state vector (a point in the Hilbert space) onto the corresponding eigenvector. George Mackey had asked whether Born's rule is a necessary consequence of a particular set of axioms for quantum mechanics, and more specifically whether every measure on the lattice of projections of a Hilbert space can be defined by a positive operator with unit trace. Though Richard Kadison proved this was false for two-dimensional Hilbert spaces, Gleason's theorem (published 1957) shows it to be true for higher dimensions. Gleason's theorem implies the nonexistence of certain types of hidden variable theories for quantum mechanics, strengthening a previous argument of John von Neumann. Von Neumann had claimed to show that hidden variable theories were impossible, but (as Grete Hermann pointed out) his demonstration made an assumption that quantum systems obeyed a form of additivity of expectation for noncommuting operators that might not hold a priori. In 1966, John Stewart Bell showed that Gleason's theorem could be used to remove this extra assumption from von Neumann's argument. Ramsey theory The Ramsey number R(k,l) is the smallest number r such that every graph with at least r vertices contains either a k-vertex clique or an l-vertex independent set. Ramsey numbers require enormous effort to compute; when max(k,l) ≥ 3 only finitely many of them are known precisely, and an exact computation of R(6,6) is believed to be out of reach. In 1953, the calculation of R(3,3) was given as a question in the Putnam Competition; in 1955, motivated by this problem, Gleason and his co-author Robert E. Greenwood made significant progress in the computation of Ramsey numbers with their proof that R(3,4) = 9, R(3,5) = 14, and R(4,4) = 18. Since then, only five more of these