The 1/3–2/3 conjecture states that, for any finite poset that is not totally ordered, there are always two elements x and y such that at least 1/3 and at most 2/3 of the linear extensions of the partial order (that is, the total orders that match the partial ordering for all comparable elements) have x < y.

Put colloquially, we have a (finite) set of n elements, and we want to order them from first to last. We can do this by picking out pairs of elements that aren't yet comparable and deciding which one we want to come before the other (and then extending this choice, together with the comparisons we have already picked, to a partial order by taking the transitive closure). The 1/3–2/3 conjecture states that, at any intermediate stage of this process, it is always possible to find two elements x and y such that, by choosing whether x should come before y or vice-versa, we reduce the number of possible orderings by at least a factor of 2/3.