Order 5: 369-380, 1989. 369 © 1989 Kluwer Academic Pubhshers. Printed m the Netherlands. Semiorders and the 1/3-2/3 Conjecture GRAHAM R. BRIGHTWELL Department of Pure Mathematics and Mathematical Statistics, UmversiO' of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England Commumcated by W T Trotter (Received: 10 June 1988; accepted: 20 August 19883 Abstracl. A well-known conjecture of Fredman is that, for every fimte partially ordered set (X. <) which is not a chain, there Js a pair of elements x..v such that P(x<y), the proportion of linear extensions of (X, <) with \ below 3', hes between 1/3 and 2/3. In this paper, we prove the conjecture in the special case when (X. <) is a semiorder. A property we call 2-separation appears to be crucial, and we classify all locally fimte 2-separated posets of bounded w~dth. AMS subject classification (1980). 06AI0. Key words. Poser, hnear extension, semtorder, 1/3-2/3 conjecture, partially ordered set. Throughout this paper, < will denote a partial order on an underlying count- able (often finite) set X. A linear extension of (X, <) is a linear ordering < of X which extends <, i.e. such that w-< y whenever x < y. We define A(<) to be the set of all linear extensions of(J(, <) and, ifX is finite, l(<) = I A(<)[. If (X, <) is finite, and F is any subset of A(<), the probability of F is just ]F I/l(<). This is, of course, equivalent to making A(<) into a probability space by setting each linear extension to be equally likely. If F is the subset of A(<) consisting of those linear extensions with x below y, for x and y elements of X, then we shall write P(x-<y) for P(F). In [1] and [2], we extended this definition of probability in A(<) to some countably infinite posets (X, <) as follows. We suppose that (X, <) is locally .finite, which we take to mean that, for every x, 3'~ X, there are only finitely many z satisfying x < z < y. We also suppose that (X, <) is thin, i.e., there is some fixed k such that every xeX is incomparable with at most k other elements in (X, <). We let (X,)( be an increasing sequence of sets whose union is X, all of which are convex in (X, <), i.e., if x and y are in Xj, and x < z < y, then also z c Xj. A subset I" of A(<) is local to Z, for Z~X, if, whenever ;teF and/1 agrees with 2 on Z, then p~F. For an event F which is local to XI, we define P(F) to be the limit as n --> oo of the probability off in the finite poser (X,,, < Ix,,). The main result of [1] is that this limit exists and is independent of the sequence (X,) chosen. Let us introduce a little more terminology. If a poset (X, <) can be par-