We prove that if P is a partial order and P → (ω)ω, then (a) P → (ω + ω + 1, 4)3, and (b) P → (ω +m,n)3 for each m,n < ω. Together these results represent the best progress known to us on the following question of P. Erdős and others. If P → (ω)ω, then does P → (α, n)3 for each α < ω1 and each n < ω?

A preference relation is a total order on a finite set and a quasipreference relation is a partial order. This paper first introduces the classic axiomatic property of preference relation, and develops a similar axiom for the quasi-preference relation.

The dualization problem on arbitrary posets is a crucial step in many applications in logics, databases, artificial intelligence and pattern mining. The objective of this paper is to study reductions of the dualization problem on arbitrary posets to the dualization problem on boolean lattices, for which output quasi-polynomial time algorithms exist. We introduce convex embedding and poset refle...

This paper presents a method to solve an ordinal classification problem where the objects are described with a set of partially ordered features. To construct an ordinal classification model we aggregate partial orders of features by weighting the incidence matrices of partial order graphs. To describe set of possible solutions we use partial order cones corresponding to the incidence matrices ...

Over the past 15 years, Ramsey theoretic techniques and concepts have been applied with great success to partially ordered sets. In the last year alone, four new applications of Ramsey theory to posets have produced solutions to some challenging combinatorial problems. First, Kierstead and Trotter showed that dimension for interval orders can be characterized by a single ramsey trail by proving...

The search for minimal elements in partially ordered sets is a generalization of the task of finding Pareto-optimal elements in multi-criteria optimization problems. Since there are usually many minimal elements within a partially ordered set, a population-based evolutionary search is, as a matter of principle, capable of finding several minimal elements in a single run and gains therefore a st...

We investigate the generalisation of quantum search of unstructured and totally ordered sets to search of partially ordered sets (posets). Two models for poset search are considered. In both models, we show that quantum algorithms can achieve at most a quadratic improvement in query complexity over classical algorithms, up to logarithmic factors; we also give quantum algorithms that almost achi...

We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting 0,∞ ,≤ that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, par...