Given a property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. We call |Pn| the speed of P. Recent research has shown that the speed of a monotone or hereditary property P can be a constant, polynomial, or exponential function of n, and the structure of the graphs in P can then be well described. Similarly, |Pn| can be of the form n(1−1/k+o(1))n or 2(1−1/k+o...

A metric space (X, d) is called c-monotone if there is a linear order < on X and c > 0 such that d(x, y) 6 c d(x, z) for all x < y < z in X. A brief account of investigation of monotone spaces including applications is presented. 1 Monotone and σ-Monotone Spaces. In [6] I investigated existence of sets in Euclidean spaces that have large Hausdorff dimension and yet host no continuous finite Bor...

A variational inequality problem (VIP) satisfying a constraint qual-iication can be reduced to a mixed complementarity problem (MCP). Monotonicity of the VIP implies that the MCP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable i...

We give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension. It is based on a Poincaré-Bendixson property enjoyed by a new class of monotone systems introduced in L. A. Sanchez, Cones of rank 2 and the PoincaréBendixson property for a new class of monotone systems, Journal of Differential Equations 216 (2009), 1170-1190. A concrete applic...

Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's " remarkable transformation. " A monotone analogue of the family of e...

Given a monotone graphical property Q, how large should d(n) be to ensure that if (Hn) is any sequence of graphs satisfying |Hn| = n and (Hn)¿d(n), then almost every induced subgraph of Hn has property Q? We prove essentially best possible results for the following monotone properties: (i) k-connected for :xed k, (ii) Hamiltonian. c © 2002 Elsevier Science B.V. All rights reserved.

Let k ≥ 2 be a fixed integer and P be a property of k-uniform hypergraphs. In other words, P is a (typically infinite) family of k-uniform hypergraphs and we say a given hypergraph H satisfies P if H ∈ P . For a given constant η > 0 a k-uniform hypergraph H on n vertices is η-far from P if no hypergraph obtained from H by changing (adding or deleting) at most ηn edges in H satisfies P . More pr...

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum of two functions with the single crossing property need not have the same property. We obtain the precise condition...