Random preorders Peter Cameron and Dudley Stark School of Mathematical Sciences Queen Mary, University of London Mile End, London, E1 4NS U.K. Submitted to Combinatorica Abstract A random preorder on n elements consists of linearly ordered equiva- lence classes called blocks. We investigate the block structure of a preorder chosen uniformly at random from all preorders on n elements as n → ∞. 1 Introduction Let R be a binary relation on a set X . We say R is reflexive if (x, x) ∈ R for all x ∈ X . We say R is transitive if (x, y) ∈ R and (y , z) ∈ R implies (x, z) ∈ R.A partial preorder is a relation R on X which is reflexive and transitive. A relation R is said to satisfy trichotomy if, for any x, y ∈ X , one of the cases (x, y) ∈ R, x = y, or (y , x) ∈ R holds. We say that R is a preorder if it is a partial preorder that satisfies trichotomy. The members of X are said to be the elements of the preorder. A relation R is antisymmetric if, whenever (x, y) ∈ R and (y , x) ∈ R both hold, then x = y. A relation R on X is a partial order if it is reflexive, transitive, and antisymmetric. A relation is a total order, if it is a partial order which satisfies trichotomy. Given a partial preorder R on X , define a new relation S on X by the rule that (x, y) ∈ S if and only if both (x, y) and (y , x) belong to R. Then S is an equivalence relation. Moreover, R induces a partial order x on the set of equivalence classes of S in a natural way: if (x, y) ∈ R, then ( x, y) ∈ R, where x is the S-equivalence class containing x and similarly for y. We will call an S- equivalence class a block. If R is a preorder, then the relation R on the equivalence 1