Ramsey Theory and Sequences of Random Variables

Abstract

We consider probability spaces which contain a family {EA[ratio ]A⊆{1, 2, …, n}, [mid ]A[mid ]=k} of events indexed by the k-element subsets of {1, 2, …, n}. A pair (A, B) of k-element subsets of {1, 2, …, n} is called a shift pair if the largest k−1 elements of A coincide with the smallest k−1 elements of B. For a shift pair (A, B), Pr[AB¯] is the probability that event EA is true and EB is false. We investigate how large the minimum value of Pr[AB¯], taken over all shift pairs, can be. As n→∞, this value converges to a number λk, with ½−1/2k+2[les ]λk[les ] ½−1/4k+2. We show that λk is a strictly increasing function of k, with λ1=¼ and λ2=1/3.For k=1, our results have the following natural interpretation. If a fair coin is tossed repeatedly, and event Ei is true when the ith toss is heads, then for all i and j with i<j, Pr[EiĒj]=¼. Furthermore, as we show in this paper, for any ε>0, there is an n such that for any sequence E1, E2, …, En of events in an arbitrary probability space, there are indices i<j with Pr[EiĒj]<¼+ε. The results and techniques we develop in this research, together with further applications of Ramsey theory, are then used to show that the supremum of fractional dimensions of interval orders is exactly 4, answering a question of Brightwell and Scheinerman.Generalizing the ¼+ε result to random variables X1, X2, …, Xn with values in an m-element set, we obtain a finite version of de Finetti's theorem without the exchangeability hypothesis: for any fixed m, k and ε, every sufficiently long sequence of such random variables has a length-k subsequence at variation distance less than ε from an i.i.d. mix.