The Hanani–Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three o...

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a gener...

This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of R, we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly’s and Carathéodory’s theorems.

Suppose that t ≥ 2 is an integer, and randomly label t graphs with the integers 1 . . . n. We give sufficient conditions for the number of edges common to all t of the labelings to be asymptotically Poisson as n → ∞. We show by example that our theorem is, in a sense, best possible. For Gn a sequence of graphs of bounded degree, each having at most n vertices, Tomescu [7] has shown that the num...

1. Introduction. A number a is called badly approximable if j a — p/q | > c/q2 for some c > 0 and all rationals pjq. It is known that an irrational number a is badly approximable if and only if the partial denominators in its continued fraction are bounded [4, Theorem 23]. In a recent paper [7] I proved results of the following type: // fuf2, • ■ • are differenliable functions whose derivatives...

The following purely topological generalization is given of Fodor’s theorem from [F1] (also known as the “pressing down lemma”): Let X be a locally compact, non-compact T2 space such that any two closed unbounded (c u b) subsets of X intersect [of course, a set is bounded if it has compact closure]; call S ⊂ X stationary if it meets every c u b in X. Then for every neighbourhood assignment U de...

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have onl...