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The "generic" family of classical sequential growth dynamics for causal sets [1] provides cosmological models of causal sets which are a testing ground for ideas about the, as yet unknown, quantum theory. In particular we can investigate how general covariance manifests itself and address the problem of identifying and interpreting covariant "observables" in quantum gravity. The problem becomes, in this setting, that of identifying measurable covariant collections of causal sets, to each of which corresponds the question: "Does the causal set that occurs belong to this collection?" It has for answer the probability measure of the collection. Answerable covariant questions, then, correspond to measurable collections of causal sets which are independent of the labelings of the causal sets. However, what the transition probabilities of the classical sequential growth dynamics provide directly is a measure on the space of labeled causal sets and the physical interpretation of the covariant measurable collections is consequently obscured. We show that there is a physically meaningful characterisation of the class of measurable covariant sets as unions and differences of "stem sets".

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A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number ∆, there is a constant r ∆ such that, for any connected n-vertex graph G with maximum degree ∆, the Ramsey number R(G, G) is at most r ∆ n, provided n is sufficiently large.In 1987, Burr made a strong conjecture implying that one may take r ∆ = ∆. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r ∆ > 2 c∆ for some constant c > 0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n) = o(n), then R(G, G) ≤ (2χ(G) + 4)n ≤ (2∆ + 6)n, i.e., r ∆ = 2∆ + 6 suffices. On the other hand, we show that Burr's conjecture itself fails even for P k n , the kth power of a path P n . Brandt showed that for any c, if ∆ is sufficiently large, there are connected nvertex graphs G with ∆(G) ≤ ∆ but R(G, K 3 ) > cn. We show that, given ∆ and H, there are β > 0 and n 0 such that, if G is a connected graph on n ≥ n 0 vertices with maximum degree at most ∆ and bandwidth at most βn, then we have R(G, H) = (χ(H) − 1)(n − 1) + σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ε(H) log n/ log log n.