Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes I. Topologies and the geodesic convexity

AbstractThe weak geodesic topology on the vertex set of a partial cube G is the finest weak topology on V(G) endowed with the geodesic convexity. We prove the equivalence of the following properties: (i) the space V(G) is compact; (ii) V(G) is weakly countably compact; (iii) the vertex set of any ray of G has a limit point; (iv) any concentrated subset of V(G) (i.e. a set A such that any two infinite subsets of A cannot be separated by deleting finitely many vertices) has a finite positive number of limit points. Moreover, if V(G) is compact, then it is scattered. We characterize the partial cubes for which the weak geodesic topology and the geodesic topology (see [N. Polat, Graphs without isometric rays and invariant subgraph properties I. J. Graph Theory27 (1998), 99–109]) coincide, and we show that the class of these particular partial cubes is closed under Cartesian products, retracts and gated amalgams

Stochastic Analysis: Geometry of Random Processes

A common feature shared by many natural objects arising in probability theory is that they tend to be very “rough”, as opposed to the “smooth” objects usually studied in other branches of mathematics. It is however still desirable to understand their geometric properties, be it from a metric, a topological, or a measure-theoretic perspective. In recent years, our understanding of such “random geometries” has seen spectacular advances on a number of fronts

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum