We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain.Keywords: Prime graph, Prime poset, The neighborhood lattice of a graph, Incidence structure, Galois lattice, Ramsey Theorem. AMS subject classification (2000). 06A06, 06A07 Presentation of the resultsThis paper is about prime graphs and prime posets. Our notations and terminology mostly follow [1]. The graphs we consider are undirected, simple and have no loops. That is, a graph is a pair G := (V, E), where E is a subset of [V ] 2 , the set of 2-element subsets of V . Elements of V are the vertices of G and elements of E its edges.

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Starting with a correspondence between prime two-dimensional orders and pairs of perpendicular total orders we put in perspective several asymptotic results, we deduce an estimate of the number of prime two-dimensional orders (labelled and unlabelled as well). Using Poisson approximation, we give a new proof of the fact that the proportion of total orders perpendicular to a given total order is asymptotically e −2 = 0.1353 . . ..

Abstract. We characterize pairs of orthogonal countable ordinals. Two ordinals α and β are orthogonal if there are two linear orders A and B on the same set V with order types α and β respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if α and β are two countable ordinals, with α ≤ β, then α and β are orthogonal if and only if either ω + 1 ≤ α or α = ω and β < ωβ.