We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin’s property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly strengthened forms, and present models with varying structures of the cofinal types of ultrafilters over measurable cardinals.

We show that for the case of uniformly convex Banach spaces, the conditions of Brøndsted’s fixed point theorem can be relaxed.

We introduce a Morse theory for posets of Bestvina–Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ over $\mathsf {RCA}_0$. For each fixed $k \geq 3$, the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$. The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$. Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$, by $\mathsf {ADS}$, and by $\mathsf {SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$.

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes of trees mentioned above.

In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.

We characterize those partially ordered sets I for which the canonical maps Mi → colim Mj into colimits of abstract sets are always injective, provided that the transition maps are injective. We also obtain some consequences for colimits of vector spaces.

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

Marques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA0 to WSAC, the corresponding principle for weakly stable posets.

The random k-dimensional partial order Pk(n) on n points is defined by taking n points uniformly at random from [0,1]k. Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.

We pay particular attention to the height Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of Hk(n) for the entire range of k(n).

We also consider the maximum degree of Pk(n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.

We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of Pk(n) for large k(n).