A probability measure on the subsets of the edge set of a graph G is a 1-independent probability measure (1-ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1-ipm , denote by G the associated random graph model. Let  1,⩾p (G) denote the collection of 1-ipms on G for which each edge is included in G with probability at least p. For G = Z 2 , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all ∈  1,⩾p (G), (G) almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p ⋆. We also determine the 1-independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f 1,G (p), the infimum over all ∈  1,⩾p (G) of the probability that G is connected. We determine f 1,G (p) exactly when G is a path, a complete graph and a cycle of length at most 5.

No abstract

We show that for any positive integer r there exists an integer k and a k-colouring of the edges of K 2 k +1 with no monochromatic odd cycle of length less than r. This makes progress on a problem of Erdős and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the k-colour Ramsey number of the odd cycle and prove that, for all odd r and all k sufficiently large, there exists a constant = (r) > 0 such that R k (C r ) > (r − 1)(2 + ) k−1 .

A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of edges that a $K_{p}$-saturated graph on $n$ vertices can have if it has minimum degree at least $t$. We prove that sat$_{t}(n,p) = tn - O(1)$, where the limit is taken as $n$ tends to infinity. This confirms a conjecture of Bollob\'as when $p = 3$. We also present constructions for graphs that give new upper bounds for sat$_{t}(n,p)$ and discuss an analogous problem for saturated hypergraphs.Comment: 15 page

No abstract