ABSTRACT:We shall consider the discrete time synchronous random majority-vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbors had at time t − 1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further more, we show that the expected time for it to reach one of these configurations from the other is (1/p

We prove that the middle two layers of the cube Q 2rþ1 contain a cycle of length ð1 À oð1ÞÞ2 2rþ1 r À Á : Our methods can also be used to show that the odd graph O k contains a cycle of length ð1 À oð1ÞÞjV ðO k Þj:One of our tools, giving a Hamilton cycle in the cube with the minimum number of 'changes of direction', may be of independent interest. r

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 .

We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from K3,3,3 has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.

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