No abstract

Zarankiewicz's Crossing Number Conjecture states that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals Z(m, n) := m/2 (m − 1)/2 n/2 (n − 1)/2 , for all positive integers m, n. This conjecture has only been verified for min{m, n} ≤ 6, for K 7,7 , K 7,8 , K 7,9 , and K 7,10 , and for K 8,8 , K 8,9 , and K 8,10 . We determine, for each positive integer m, an integer N 0 = N 0 (m) with the following property: if cr(Km,n) = Z(m, n) for all n ≤ N 0 , then cr(Km,n) = Z(m, n) for every n. This yields, for each fixed integer m, a finite algorithm that either proves that cr(Km,n) = Z(m, n) for every n, or else finds a counterexample.

We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite.

Abstract. We show that every additively representable comparative probability order on n atoms is determined by at least n − 1 binary subset comparisons. We show that there are many orders of this kind, not just the lexicographic order. These results provide answers to two questions of Fishburn et al (2002). We also study the flip relation on the class of all comparative probability orders introduced by Maclagan. We generalise an important theorem of Fishburn, Pekeč and Reeds, by showing that in any minimal set of comparisons that determine a comparative probability order, all comparisons are flippable.By calculating the characteristics of the flip relation for n = 6 we discover that the polytopes associated with the regions in the corresponding hyperplane arrangement can have no more than 13 facets and that there are 20 regions whose associated polytopes have 13 facets. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable.

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).