Let Δ > 1 be a fixed positive integer. For \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\textbf{ {z}}} \in \mathbb{R}_+^\Delta\end{align*} \end{document} let Gz be chosen uniformly at random from the collection of graphs on ∥z∥1n vertices that have zin vertices of degree i for i = 1,…,Δ. We determine the likely evolution in continuous time of the SIR model for the spread of an infectious disease on Gz, starting from a single infected node. Either the disease halts after infecting only a small number of nodes, or an epidemic spreads to infect a linear number of nodes. Conditioning on the event that more than a small number of nodes are infected, the epidemic is likely to follow a trajectory given by the solution of an associated system of ordinary differential equations. These results also give the likely number of nodes infected during the course of the epidemic and the likely length in time of the epidemic. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C 4 . We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((n log n) 1/3 ) and consequently size O(n 4/3 log(n) 1/3 ), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollobás

denote the diamond graph, formed by removing an edge from the complete graph K 4 . We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of K − 4 . We show that, with probability tending to 1 as n → ∞, the final size of the graph produced is ( log n · n 3/2 ). Our analysis also suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random-looking subgraph.

A triangle T (r) in an r-uniform hypergraph is a set of r + 1 edges such that r of them share a common (r − 1)-set of vertices and the last edge contains the remaining vertex from each of the first r edges. Our main result is that the random greedy triangle-free process on n points terminates in an r-uniform hypergraph with independence number O((n log n) 1/r ). As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number r(T (r) , K (r) s ) has order of magnitude s r / log s. This answers questions posed in [4, 10] and generalizes the celebrated results of Ajtai-Komlós-Szemerédi [1] and Kim [9] to hypergraphs.The upper bound was proved by Ajtai-Komlós-Szemerédi [1] as one of the first applications of the semi-random method in combinatorics (simpler proofs now exist due to Shearer [12,13]). The lower bound, due to Kim [9], was also achieved by using the semi-random or nibble method. More recently, the first author [3] showed that a lower bound for r(K 3 , K s ) could also be obtained by the triangle-free process, which is a random greedy algorithm. This settled a question of Spencer on the independence number of the triangle-free process. Still more recently, Bohman-Keevash [6] and Fiz Pontiveros-Griffiths-Morris [8] have analyzed the triangle-free process more carefully and improved the constants obtained so that the gap between the upper and lower bounds for r(K 3 , K s ) is now asymptotically a multiplicative factor of 4.Given the difficulty of these basic questions in graph Ramsey theory, one would expect that the corresponding questions for hypergraphs are hopeless. This is not always the case. Hypergraphs

The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete r-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour, Akbari and Ebrahimi [2].