See [1] for proofs and terminology.

Covering graphs of various types of finite partially ordered sets have been extensively studied since Ore in [2] asked for a characterization of such graphs in general. In this note we introduce a graph, called the m-chain graph of a poset, that reduces to the covering graph when m = 1. Since we will show how to construct the m-chain graph for m > 2 as the 2-chain graph of another poset, we are...

̂ M of the connection ∇M to the universal cover ̂ M of M is a locally flat connection. The affine structure of ̂ M is defined by a local diffeomorphism DM : ̂ M → R called the developing map. The developing map gives rise to a representation hM : π1 M → Aff R called the holonomy. The linear part L hM of hM is the linear holonomy. The affine manifold M,∇M is complete if and only if DM is a diffeomor...

In this paper, we address a particular variation of the Turán problem for the hypercube. Alon, Krech and Szabó (2007) asked “In an n-dimensional hypercube, Qn, and for ` < d < n, what is the size of a smallest set, S, of Q`’s so that every Qd contains at least one member of S?” Likewise, they asked a similar Ramsey type question: “What is the largest number of colors that we can use to color th...

We describe a general framework for covering concepts using terminologies and briefly present the already investigated instances of this framework. Then, we formalize the best covering problem in the context of the ALN language, in which the difference operation is not semantically unique, and sketches the technique to solve the underlying computation problems.

In the paper we prove a factorization theorem for representations of fundamental groups of compact Kähler manifolds (Kähler groups) into solvable matrix groups. We apply this result to prove that the universal covering of a compact Kähler manifold with a residually solvable fundamental group is holomorphically convex.

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p : (Q̃, q̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q) := Aut(p) = {g ∈ Adj(Q)′ | qg = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected c...