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.