A stochastic Ramsey theorem

We establish a stochastic extension of Ramseyʹs theorem. Any Markov chain generates a filtration relative to which one may define a notion of stopping times. A stochastic colouring is any k-valued ( k < ∞ ) colour function defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time θ and any infinite history ω of the Markov chain, let ω | θ denote the finite partial history up to and including the time θ ( ω ) . Given k = 2 , for every ϵ > 0 , we prove that there is an increasing sequence θ 1 < θ 2 < ⋯ of bounded stopping times having the property that, with probability greater than 1 − ϵ , the history ω is such that the values assigned to all pairs ( ω | θ i , θ j ) , with i < j , are the same. Just as with the classical Ramsey theorem, we also obtain an analogous finitary stochastic Ramsey theorem. Furthermore, with appropriate finiteness assumptions, the time one must wait for the last stopping time (in the finitary case) is uniformly bounded, independently of the probability transitions. We generalise the results to any finite number k of colours.