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We generalize Reimer's Inequality [6] (a.k.a. the BKR Inequality or the van den Berg–Kesten Conjecture [1]) to the setting of finite distributive lattices.
Let be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U() be the set of truth assignments that satisfy exactly one term in . Motivated by questions in computational complexity, Rudich conjectured that there exist ∊, δ > 0 such that, if is any set of terms for which U() contains at least a (1−∊)-fraction of all truth assignments, then there exists a term t ∈ such that at least a δ-fraction of assignments satisfy some term of sharing a variable with t [8].
We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U() is at least 1 − ∊, there exists a t ∈ such that the measure of the set of assignments satisfying either t or some term incompatible with t (i.e., having no satisfying assignments in common with t) is at least . (A key part of the proof is a correlation-like inequality on events in a finite product probability space that is in some sense dual to Reimer's inequality [11], a.k.a. the BKR inequality [5], or the van den Berg–Kesten conjecture [3].)
Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X]n of unordered n-tuples from X, find a subset Y ⊆ X such that all elements of [Y]n get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets.
Polychromatic Ramsey theory is concerned with a “dual” problem, in which we are given a colouring of [X]n and are looking for subsets Y ⊆ X such that any two distinct elements of [Y]n get different colours. Subsets with this property are called polychromatic or rainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integer k we say that a colouring is k-bounded when each colour is used for at most k many n-tuples.
At this point it will be convenient to introduce a compact notation for stating results in polychromatic Ramsey theory. We recall that in classical Ramsey theory we write to mean “every colouring of [κ]n in k colours has a monochromatic set of order type α”. We will write to mean “every k-bounded colouring of [κ]n has a polychromatic set of order type α”. We note that when κ is infinite and k is finite a k-bounded colouring will use exactly κ colours, so we may as well assume that κ is the set of colours used.
Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic.
We show that almost surely , where G is a random tree of order n or the giant component of a random graph with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.
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