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Consider the following stochastic graph process. We begin with G 0 , the empty graph on n vertices, and form G i by adding a randomly chosen edge e i to G i−1 where e i is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in G i−1 nor form triangles when added as edges to G i−1. Let the random variable M be the(More)
Let c be a constant and (e1, f1), (e2, f2),. .. , (ecn, fcn) be a sequence of ordered pairs of edges on vertex set [n] chosen uniformly and independently at random. Let A be an algorithm for the on-line choice of one edge from each presented pair, and for i = 1,. .. , cn let GA(i) be the graph on vertex set [n] consisting of the first i edges chosen by A.(More)
The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we(More)
Let e1, e2,. .. be a sequence of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). Our goal is to choose, for m as large as possible, a subset such that the size of the largest component in G = ([n], E) is o(n) (i.e. G does not contain a giant component). Furthermore, the selection process must(More)
Let c be a constant and a sequence of ordered pairs of edges from the complete graph K n chosen uniformly and independently at random. We prove that there exists a constant c 2 such that if c > c 2 then whp every graph which contains at least one edge from each ordered pair (e i , f i) has a component of size Ω(n) and if c < c 2 then whp there is a graph(More)
Let G 3−out denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G 3−out has minimum degree 3 and average degree 6. We prove that the probability that G 3−out is Hamiltonian goes to 1 as n tends to infinity.
A set S of positive integers has distinct subset sums if the set x∈X x : X ⊂ S has 2 |S| distinct elements. Let f (n) = min{max S : |S| = n and S has distinct subset sums}. In 1931 Paul Erd˝ os conjectured that f (n) ≥ c2 n for some constant c. In 1967 John Conway and Richard Guy constructed an interesting sequence of sets of integers. They conjectured that(More)