Abstract. We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability min(1, κ(X i , X j ) log n/n), where κ ≥ 0 is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.

We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius r that determines the "visible neighbors" of each node and the number of edges c that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.

Abstract. Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs. 1

One of the De Bruijn -Erdős theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, nearpencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u, v, w} is a hyperedge. With this definition, the De Bruijn -Erdős theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case. 1