A combinatorial theorem

A Combinatorial Theorem for Trees

A functional combinatorial central limit theorem

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein’s met...

A Combinatorial Theorem on Systems of Sets

It is quite easy to see that (2) and (3) give best possible results. Upper estimates for n were also given in [2] if either (Alt..., An)eS(k, </, m) and /<^(m+l) or if (Alt..., An)eS(k, I, m) and l^^(m+k), but the results in these cases are not best possible. Other results of this kind have been established in [3] and [4] which settle certain conjectures made in [2]. In this note we establish t...

A Combinatorial Version of Vaaler’s Theorem

In 1979 Vaaler proved that every d-dimensional central section of the cube [−1, 1]n has volume at least 2d. We prove the following sharp combinatorial analogue. Let K be a d-dimensional subspace of Rn. Then, there is a probability measure P on the section [−1, 1]n ∩K, so that the quadratic form∫ [−1,1]n∩K v ⊗ v dP (v) dominates the identity on K (in the sense that the difference is positive sem...

A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

Given a nonnegative integer m and a finite collection A of linear forms on Qd, the arrangement of affine hyperplanes in Qd defined by the equations α(x) = k for α ∈ A and integers k ∈ [−m,m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of Am are quasi-polynomials inm and that they satisfy a simple combinatorial reciprocity law.